School Subjects: Mathematics

Operation of Sets

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Introduction to Sets</h1>
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Forget everything you know about numbers.</div>
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In fact, forget you even know what a number is.</div>
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This is where mathematics starts.</div>
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Instead of math with numbers, we will now think about math with "things".</div>
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Definition</h2>
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What is a set? Well, simply put, it's <b>a collection</b>.</div>
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First we specify a common property among "things" (this word will be defined later) and then we gather up all the "things" that have this common property.</div>
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<img height="160" src="http://www.mathsisfun.com/sets/images/set-clothes.jpg" width="200" /></div>
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For example, the items you wear: shoes, socks, hat, shirt, pants, and so on.</div>
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I'm sure you could come up with at least a hundred.</div>
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This is known as a <b>set</b>.</div>
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Or another example is <b>types of fingers</b>.</div>
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This set includes index, middle, ring, and pinky.</div>
</td><td style="line-height: 22px;"></td><td style="line-height: 22px;"><img height="96" src="http://www.mathsisfun.com/sets/images/set-fingers.jpg" width="120" /></td></tr>
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So it is just things grouped together with a certain property in common.</div>
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Notation</h2>
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There is a fairly simple notation for sets. We simply list each element, separated by a comma, and then put some curly brackets around the whole thing.</div>
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<img alt="Set" height="127" src="http://www.mathsisfun.com/sets/images/set.gif" width="381" /></div>
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The curly brackets <span class="large" style="color: #a06000; font-size: 19px;">{ }</span> are sometimes called "set brackets" or "braces".</div>
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This is the notation for the two previous examples:</div>
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<b>{socks, shoes, watches, shirts, ...}</b><br />
<b>{index, middle, ring, pinky}</b></div>
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Notice how the first example has the "..." (three dots together).</div>
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The three dots <span class="huge" style="font-size: 35px; font-weight: bold;">...</span> are called an ellipsis, and mean "continue on".</div>
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So that means the first example continues on ... for infinity.</div>
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<i>(OK, there isn't <b>really</b> an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)</i></div>
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So:</div>
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<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">The first set <b>{socks, shoes, watches, shirts, ...}</b> we call an <b>infinite set</b>,</li>
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">the second set <b>{index, middle, ring, pinky}</b> we call a <b>finite set</b>.</li>
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But sometimes the "..." can be used in the middle to save writing long lists:</div>
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Example: the set of letters:</div>
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<b>{a, b, c, ..., x, y, z}</b></div>
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In this case it is a <b>finite set</b> (there are only 26 letters, right?)</div>
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Numerical Sets</h2>
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So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?</div>
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Set of even numbers: {..., -4, -2, 0, 2, 4, ...}<br />
Set of odd numbers: {..., -3, -1, 1, 3, ...}<br />
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}<br />
Positive multiples of 3 that are less than 10: {3, 6, 9}</div>
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And the list goes on. We can come up with all different types of sets.</div>
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There can also be sets of numbers that have no common property, they are just <b>defined</b> that way. For example:</div>
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{2, 3, 6, 828, 3839, 8827}<br />
{4, 5, 6, 10, 21}<br />
{2, 949, 48282, 42882959, 119484203}</div>
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Are all sets that I just randomly banged on my keyboard to produce.</div>
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Why are Sets Important?</h2>
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Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.</div>
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<i>Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: <b>Sets</b></i>.</div>
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Universal Set</h2>
<table border="0" style="color: black; font-family: Verdana, Arial, Tahoma, sans-serif; font-size: 15px; line-height: 23px;"><tbody>
<tr><td style="line-height: 22px;"><img height="74" src="http://www.mathsisfun.com/sets/images/star.jpg" width="99" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">At the start we used the word "things" in quotes. We call this the <b>universal set</b>. It's a set that contains everything. Well, not <i>exactly</i> everything. <b>Everything that is relevant to our question.</b></td></tr>
<tr><td style="line-height: 22px;"><img height="80" src="http://www.mathsisfun.com/sets/images/various-integers.gif" width="100" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">Then our sets included integers. The universal set for that would be all the integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers.</td></tr>
<tr><td style="line-height: 22px;"><img height="80" src="http://www.mathsisfun.com/sets/images/various-reals.gif" width="100" /></td><td style="line-height: 22px;"></td><td style="line-height: 22px;">However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.</td></tr>
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Some More Notation</h2>
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<table align="center" border="0" style="background-color: #cceeff; background-position: initial initial; background-repeat: initial initial; border-collapse: collapse; width: 93%px;"><tbody>
<tr><td style="border: 2px solid white; font-size: 15px; line-height: 22px; padding: 5px;"><img height="80" src="http://www.mathsisfun.com/sets/images/aa.gif" width="60" /></td><td style="border: 2px solid white; font-size: 15px; line-height: 22px; padding: 5px;">When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set.<br />
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So for example, A is a set, and a is an element in A. Same with B and b, and C and c.</td></tr>
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Now you don't have to listen to the standard, you can use something like <i><b>m</b></i> to represent a set without breaking any mathematical laws (watch out, you can get <b><span style="font-family: Times New Roman, Times, serif;">π</span></b> years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?</div>
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Also, when we say an element <b>a</b> is in a set <b>A</b>, we use the symbol <img height="12" src="http://www.mathsisfun.com/sets/images/element-symbol.png" width="12" /> to show it.<br />
And if something is not in a set use <img height="14" src="http://www.mathsisfun.com/sets/images/not-element-symbol.png" width="12" />.</div>
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Example: Set <b>A</b> is {1,2,3}. We can see that <b>1 <img height="12" src="http://www.mathsisfun.com/sets/images/element-symbol.png" width="12" /> A</b>, but <b>5 <img height="14" src="http://www.mathsisfun.com/sets/images/not-element-symbol.png" width="12" /> A</b></div>
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Equality</h2>
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Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!</div>
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Example: Are A and B equal where:</div>
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<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">A is the set whose members are the first four positive whole numbers</li>
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">B = {4, 2, 1, 3}</li>
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Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: <b>Yes, they are equal!</b></div>
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And the equals sign (=) is used to show equality, so we write:</div>
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A = B</div>
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Subsets</h2>
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When we define a set, if we take pieces of that set, we can form what is called a <b>subset</b>.</div>
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So for example, we have the set {1, 2, 3, 4, 5}. A <b>subset</b> of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:</div>
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A is a subset of B if and only if every element of A is in B.</div>
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So let's use this definition in some examples.</div>
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Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?</h3>
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1 is in A, and 1 is in B as well. So far so good.</div>
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3 is in A and 3 is also in B.</div>
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4 is in A, and 4 is in B.</div>
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That's all the elements of A, and every single one is in B, so we're done.</div>
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Yes, A is a subset of B</div>
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Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.</div>
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Let's try a harder example.</div>
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Example: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A?</h3>
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Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.</div>
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The sets are:</div>
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<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">A = {..., -8, -4, 0, 4, 8, ...}</li>
<li style="color: #000088; line-height: 24px; list-style-type: disc; padding: 0px 0px 5px;">B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}</li>
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By pairing off members of the two sets, we can see that every member of A is also a member of B, but every member of B is not a member of A:</div>
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<br /></div>
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<img alt="pairing off A and B" height="91" src="http://www.mathsisfun.com/sets/images/sets-example2.gif" width="357" /></div>
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So:</div>
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A is a subset of B, but B is not a subset of A</div>
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Proper Subsets</h2>
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If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.</div>
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Let <b>A</b> be a set. Is every element<b> </b>in <b>A</b> an element in <b>A</b>? (Yes, I wrote that correctly.)</div>
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Well, umm, <i>yes of course</i>, right?</div>
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So doesn't that mean that <b><i>A is a subset of A</i></b>?</div>
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This doesn't seem very <i>proper</i>, does it? We want our subsets to be <i>proper</i>. So we introduce (what else but) <b>proper subsets</b>.</div>
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A is a <b>proper</b> subset of B if and only if every element in A is also in B, and there exists <b>at least one element</b> in B that is <b>not</b> in A.</div>
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This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.</div>
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Example:</h3>
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{1, 2, 3} is a <b>subset</b> of {1, 2, 3}, but is <b>not </b>a<b> proper subset</b> of {1, 2, 3}.</div>
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Example:</h3>
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{1, 2, 3} <b>is </b>a<b> proper subset</b> of {1, 2, 3, 4} because the element 4 is not in the first set.</div>
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Notice that if A is a proper subset of B, then it is also a subset of B.</div>
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Even More Notation</h2>
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When we say that A is a subset of B, we write A <img height="16" src="http://www.mathsisfun.com/sets/images/subset-symbol.png" width="16" /> B.</div>
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Or we can say that A is not a subset of B by A <img height="16" src="http://www.mathsisfun.com/sets/images/not-subset-symbol.png" width="16" /> B ("A is not a subset of B")</div>
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When we talk about proper subsets, we take out the line underneath and so it becomes A <img height="12" src="http://www.mathsisfun.com/sets/images/proper-subset-symbol.png" width="15" /> B or if we want to say the opposite, A <img height="13" src="http://www.mathsisfun.com/sets/images/not-proper-subset-symbol.png" width="16" /> B.</div>
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Empty (or Null) Set</h2>
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This is probably the weirdest thing about sets.</div>
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<img height="180" src="http://www.mathsisfun.com/sets/images/keys-on-guitar.jpg" width="180" /></div>
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As an example, think of the set of piano keys on a guitar.</div>
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<i>"But wait!" you say, "There are no piano keys on a guitar!"</i></div>
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And right you are. It is a set with <b>no elements</b>.</div>
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This is known as the <b>Empty Set</b> (or Null Set).There aren't any elements in it. Not one. Zero.</div>
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It is represented by <img height="13" src="http://www.mathsisfun.com/sets/images/null-set.png" width="16" /></div>
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Or by <b>{}</b> (a set with no elements)</div>
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Some other examples of the empty set are <b>the set of countries south of the south pole</b>.</div>
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So what's so weird about the empty set? Well, that part comes next.</div>
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Empty Set and Subsets</h2>
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So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. <i><b>Is the empty set a subset of A?</b></i></div>
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Going back to our definition of subsets, <b>if every element in the empty set is also in A, then the empty set is a subset of A</b>. But what if we have <b>no</b> elements?</div>
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It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.</div>
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A good way to think about it is: <i>we can't find any elements in the empty set that aren't in A</i>, so it must be that all elements in the empty set are in A.</div>
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So the answer to the posed question is a resounding <b>yes</b>.</div>
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The empty set is a subset of every set, including the empty set itself.</div>
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Order</h2>
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No, not the order of the elements. In sets it <b>does not matter what order the elements are in</b>.</div>
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Example: {1,2,3,4) is the same set as {3,1,4,2}</div>
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When we say "order" in sets we mean <b>the size of the set</b>.</div>
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Just as there are finite and infinite sets, each has finite and infinite order.</div>
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For finite sets, we represent the order by a number, the number of elements.</div>
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Example, {10, 20, 30, 40} has an order of 4.</div>
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For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.</div>

Operation of Sets

Introduction to Sets Forget everything you know about numbers. In fact, forget you even know what a number is. This is where mathema...

Some Definations College Algebra

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<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">ALGEBRAIC EXPRESSION</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is an </span><a href="http://en.wikipedia.org/wiki/Expression_(mathematics)" title="Expression (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">expression</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> built up from </span><a href="http://en.wikipedia.org/wiki/Constant_(mathematics)" title="Constant (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">constants</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">variables</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, and the </span><a href="http://en.wikipedia.org/wiki/Algebraic_operation" title="Algebraic operation"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">algebraic operations</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> (</span><a href="http://en.wikipedia.org/wiki/Addition" title="Addition"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">addition</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Subtraction" title="Subtraction"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">subtraction</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">multiplication</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Division_(mathematics)" title="Division (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">division</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> and</span><a href="http://en.wikipedia.org/wiki/Exponentiation" title="Exponentiation"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">exponentiation</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> by an exponent that is
a </span><a href="http://en.wikipedia.org/wiki/Rational_number" title="Rational number"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">rational
number</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">).<o:p></o:p></span></div>
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<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">POLYNOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is an </span><a href="http://en.wikipedia.org/wiki/Mathematical_expression" title="Mathematical expression"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">expression</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> consisting of </span><a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">variables</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> (or <a href="http://en.wikipedia.org/wiki/Indeterminate_(variable)" title="Indeterminate (variable)"><span style="color: black; mso-themecolor: text1;">indeterminates</span></a>)
and </span><a href="http://en.wikipedia.org/wiki/Coefficient" title="Coefficient"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">coefficients</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, that involves only the
operations of</span><a href="http://en.wikipedia.org/wiki/Addition" title="Addition"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">addition</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Subtraction" title="Subtraction"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">subtraction</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, </span><a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">multiplication</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, and non-negative </span><a href="http://en.wikipedia.org/wiki/Integer" title="Integer"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">integer</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> </span><a href="http://en.wikipedia.org/wiki/Exponent" title="Exponent"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">exponents</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">. An example of a polynomial
of a single indeterminate (or variable), x, isx2 − 4x + 7, which
is a </span><a href="http://en.wikipedia.org/wiki/Quadratic_function" title="Quadratic function"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">quadratic polynomial</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.<o:p></o:p></span></div>
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<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A MONOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a product of powers of </span><a href="http://en.wikipedia.org/wiki/Variable_(mathematics)" title="Variable (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">variables</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> with </span><a href="http://en.wikipedia.org/wiki/Nonnegative_integer" title="Nonnegative integer"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">nonnegative integer</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> exponents, or, in other words, a product of variables, possibly
with repetitions.</span></div>
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<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A BINOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a </span><a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomial</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> which is the sum of two
terms, which are </span><a href="http://en.wikipedia.org/wiki/Monomial" title="Monomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">monomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.</span><a href="http://en.wikipedia.org/wiki/Binomial#cite_note-1"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">[1]</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> It is the simplest kind
of polynomial after the </span><a href="http://en.wikipedia.org/wiki/Monomial" title="Monomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">monomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.<o:p></o:p></span></div>
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<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A TRINOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a </span><a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomial</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> consisting of three
terms or </span><a href="http://en.wikipedia.org/wiki/Monomial" title="Monomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">monomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">.<o:p></o:p></span></div>
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<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A MULTINOMIAL</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is an algebraic expression having more than one term. For example:
9x3 + 2x2 + 5<br />
<b>IN ALGEBRA</b> a term is either a single
number or variable, or numbers and variables multiplied together. Terms are
separated by + or – signs<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">A VARIABLE</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a symbol that denotes a mathematical object, which could be
a </span><a href="http://en.wikipedia.org/wiki/Number" title="Number"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">number</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">, a </span><a href="http://en.wikipedia.org/wiki/Vector_(mathematics)" title="Vector (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">vector</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">,
a </span><a href="http://en.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics)"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">matrix</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">,
or even a function. In this case, the original property of
"variability" of a variable is not kept (except, sometimes, for
informal explanations).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<a href="http://www.math.com/school/subject2/lessons/S2U2L4GL.html"><b><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">LIKE TERMS</span></b></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> are terms that contain the same variables raised to the same
power. Only the </span><a href="http://www.math.com/school/subject2/lessons/S2U2L4GL.html"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">coefficients</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> of like terms are
different.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">UNLIKE TERMS</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> : The terms having the same variable with different exponents or
different variable with same exponents are calledUnlike terms .<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">EXPONENTS</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> are shorthand for repeated multiplication of the same thing by itself.
For instance, the shorthand for multiplying three copies of the
number 5 is shown on the right-hand side of the "equals"
sign in (5)(5)(5) = 53. The "exponent", being 3 in
this example, stands for however many times the value is being multiplied. The
thing that's being multiplied, being5 in this example, is called the
"base".<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">DEFINITION OF NUMERICAL COEFFICIENT -</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">The constant multiplicative factors attached
to the variables in an expression are known as Numerical Coefficient.<b><o:p></o:p></b></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">FACTORS</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">
are numbers you can multiply together to get another number:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">THE PRODUCT RULE</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a formal rule for differentiating problems where one function is
multiplied by another.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">THE QUOTIENT RULE</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a formal rule for differentiating problems where one function is
divided by another.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> <b>THE POWER RULE</b> is one of the most important </span><a href="http://en.wikipedia.org/wiki/Differentiation_rules" title="Differentiation rules"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">differentiation rules</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">. Since differentiation is linear, </span><a href="http://en.wikipedia.org/wiki/Polynomial" title="Polynomial"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomials</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> can be differentiated
using this rule.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;">SYNTHETIC DIVISION</span></b><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> is a shorthand, or shortcut, method of </span><a href="http://www.purplemath.com/modules/polydiv.htm"><span style="color: black; font-family: "Comic Sans MS"; font-size: 10.0pt; line-height: 115%; mso-bidi-font-size: 11.0pt; mso-themecolor: text1;">polynomial division</span></a><span style="font-family: 'Comic Sans MS'; font-size: 10pt; line-height: 115%;"> in the special case of
dividing by a linear factor -- and it only works in this case.
Synthetic division is generally used, however, not for dividing out factors but
for finding zeroes (or roots) of polynomials. More about this later.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<h3 style="background: white; margin-bottom: 2.25pt; margin-left: 0in; margin-right: 0in; margin-top: 11.25pt;">
<span style="font-family: Arial, sans-serif; font-size: 17pt; font-weight: normal; line-height: 115%;">Exponents rules and properties<o:p></o:p></span></h3>
<table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="background: white; border-collapse: collapse; border: none; mso-border-alt: solid #CCCCCC .75pt; mso-padding-alt: 3.75pt 3.75pt 3.75pt 3.75pt; mso-yfti-tbllook: 1184; width: 620px;">
 <tbody>
<tr>
  <td style="background: #E0E0E0; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div align="center" class="MsoNormal" style="margin-bottom: 3.75pt; text-align: center;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Rule name<o:p></o:p></span></div>
</td>
  <td style="background: #E0E0E0; border-left: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div align="center" class="MsoNormal" style="margin-bottom: 3.75pt; text-align: center;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Rule<o:p></o:p></span></div>
</td>
  <td style="background: #E0E0E0; border-left: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div align="center" class="MsoNormal" style="margin-bottom: 3.75pt; text-align: center;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Example<o:p></o:p></span></div>
</td>
 </tr>
<tr>
  <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#product"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Product rules</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">·<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"><sup> </sup></span><sup>m</sup></i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"><sup> </sup></span><sup>n+m</sup></i><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">2<sup>3</sup><span class="apple-converted-space"> </span>· 2<sup>4</sup><span class="apple-converted-space"> </span>= 2<sup>3+4</sup><span class="apple-converted-space"> </span>= 128<o:p></o:p></span></div>
</td>
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<tr>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">·<span class="apple-converted-space"> </span><i>b<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></i><span class="apple-converted-space"> </span>= (<i>a<span class="apple-converted-space"> </span></i>·<span class="apple-converted-space"><i> </i></span><i>b</i>)<span class="apple-converted-space"><i><sup> </sup></i></span><i><sup>n</sup></i><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">3<sup>2</sup><span class="apple-converted-space"> </span>· 4<sup>2</sup><span class="apple-converted-space"> </span>= (3·4)<sup>2</sup><span class="apple-converted-space"> </span>= 144<o:p></o:p></span></div>
</td>
 </tr>
<tr>
  <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#quotient"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Quotient rules</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">/<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"><sup> </sup></span><sup>m</sup></i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>a<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></i><sup>-<i>m</i></sup><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">2<sup>5</sup><span class="apple-converted-space"> </span>/ 2<sup>3</sup><span class="apple-converted-space"> </span>= 2<sup>5-3</sup><span class="apple-converted-space"> </span>= 4<o:p></o:p></span></div>
</td>
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<tr>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">a<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">/<span class="apple-converted-space"> </span><i>b<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></i><span class="apple-converted-space"> </span>= (<i>a<span class="apple-converted-space"> </span></i>/<span class="apple-converted-space"><i> </i></span><i>b</i>)<span class="apple-converted-space"><i><sup> </sup></i></span><i><sup>n</sup></i><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">4<sup>3</sup><span class="apple-converted-space"> </span>/ 2<sup>3</sup><span class="apple-converted-space"> </span>= (4/2)<sup>3</sup><span class="apple-converted-space"> </span>= 8<o:p></o:p></span></div>
</td>
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<tr>
  <td rowspan="4" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#power"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Power rules</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(<i>b<sup>n</sup></i>)<i><sup>m</sup></i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>b<sup>n·m</sup></i><o:p></o:p></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(2<sup>3</sup>)<sup>2</sup><span class="apple-converted-space"> </span>= 2<sup>3·2</sup><span class="apple-converted-space"> </span>= 64<o:p></o:p></span></div>
</td>
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<tr>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><sub><span style="font-size: 17pt; line-height: 115%;">b</span></sub></i><i><span style="font-size: 13.5pt; line-height: 115%;">n<sup>m</sup></span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><sub><span style="font-size: 17pt; line-height: 115%;">=
  b</span></sub><span style="font-size: 17pt; line-height: 115%;">(</span><i><span style="font-size: 13.5pt; line-height: 115%;">n<sup>m</sup></span></i><span style="font-size: 17pt; line-height: 115%;">)<o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><sub><span style="font-size: 17pt; line-height: 115%;">2</span></sub><span style="font-size: 13.5pt; line-height: 115%;">3<sup>2</sup></span><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><sub><span style="font-size: 17pt; line-height: 115%;">=
  2</span></sub><span style="font-size: 17pt; line-height: 115%;">(</span><span style="font-size: 13.5pt; line-height: 115%;">3<sup>2</sup></span><span style="font-size: 17pt; line-height: 115%;">)<sub>= 512</sub><o:p></o:p></span></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><sup><span style="font-size: 17pt; line-height: 115%;">m</span></sup></i><span style="font-size: 17pt; line-height: 115%;">√(<i>b<sup>n</sup></i>)
  =<span class="apple-converted-space"> </span><i>b</i><span class="apple-converted-space"> </span><i><sup>n</sup></i><sup>/<i>m</i></sup><o:p></o:p></span></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><sup><span style="font-size: 17pt; line-height: 115%;">2</span></sup><span style="font-size: 17pt; line-height: 115%;">√(2<sup>6</sup>)
  = 2<sup>6/2</sup><span class="apple-converted-space"> </span>= 8<o:p></o:p></span></span></div>
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<tr>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b</span></i><sup><span style="font-size: 17pt; line-height: 115%;">1/<i>n</i></span></sup><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">=<span class="apple-converted-space"> </span><i><sup>n</sup></i>√<i>b</i><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">8<sup>1/3</sup><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><sup>3</sup>√8<span class="apple-converted-space"> </span>= 2<o:p></o:p></span></div>
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 </tr>
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  <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><a href="http://www.rapidtables.com/math/number/exponent.htm#negative"><span style="color: black; font-family: "Arial","sans-serif"; font-size: 10.5pt; line-height: 115%; mso-themecolor: text1;">Negative exponents</span></a><span style="font-size: 10.5pt; line-height: 115%;"><o:p></o:p></span></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b<sup>-n</sup></span></i><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">= 1 /<span class="apple-converted-space"> </span><i>b<sup>n</sup></i><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">2<sup>-3</sup><span class="apple-converted-space"> </span>= 1/2<sup>3</sup><span class="apple-converted-space"> </span>= 0.125<o:p></o:p></span></div>
</td>
 </tr>
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  <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Zero rules<o:p></o:p></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b</span></i><sup><span style="font-size: 17pt; line-height: 115%;">0</span></sup><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">= 1<o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">5<sup>0</sup><span class="apple-converted-space"> </span>= 1<o:p></o:p></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">0<i><sup>n</sup></i><span class="apple-converted-space"> </span>= 0 , for<span class="apple-converted-space"> </span><i>n</i>>0<o:p></o:p></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">0<sup>5</sup><span class="apple-converted-space"> </span>= 0<o:p></o:p></span></div>
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  <td rowspan="2" style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">One rules<o:p></o:p></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><i><span style="font-size: 17pt; line-height: 115%;">b</span></i><sup><span style="font-size: 17pt; line-height: 115%;">1</span></sup><span class="apple-converted-space"><span style="font-size: 17pt; line-height: 115%;"> </span></span><span style="font-size: 17pt; line-height: 115%;">=<span class="apple-converted-space"> </span><i>b</i><o:p></o:p></span></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">5<sup>1</sup><span class="apple-converted-space"> </span>= 5<o:p></o:p></span></div>
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<tr>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">1<i><sup>n</sup></i><span class="apple-converted-space"> </span>= 1<o:p></o:p></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">1<sup>5</sup><span class="apple-converted-space"> </span>= 1<o:p></o:p></span></div>
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  <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Minus one rule<o:p></o:p></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit;"><span style="font-size: 17pt; line-height: 115%;"><img alt="http://www.rapidtables.com/math/number/exponent/minus_1_rule.gif" border="0" height="53" src="file:///C:/Users/STATIO~1/AppData/Local/Temp/msohtmlclip1/01/clip_image001.gif" v:shapes="Picture_x0020_1" width="210" /></span><span style="font-size: 17pt; line-height: 115%;"><o:p></o:p></span></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(-1)<sup>5</sup><span class="apple-converted-space"> </span>= -1<o:p></o:p></span></div>
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<tr>
  <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Derivative rule<o:p></o:p></span></div>
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  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(<i>x<sup>n</sup></i>)<i>'<span class="apple-converted-space"> </span></i>=<span class="apple-converted-space"> </span><i>n</i>·<i>x<span class="apple-converted-space"><sup> </sup></span><sup>n</sup></i><sup>-1</sup><o:p></o:p></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">(<i>x</i><sup>3</sup>)<i>'<span class="apple-converted-space"> </span></i>= 3·<i>x</i><sup>3-1</sup><o:p></o:p></span></div>
</td>
 </tr>
<tr>
  <td style="border-top: none; border: solid #CCCCCC 1.0pt; mso-border-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 10.5pt; line-height: 115%;">Integral rule<o:p></o:p></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">∫<span class="apple-converted-space"><i> </i></span><i>x<sup>n</sup>dx</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>x<sup>n</sup></i><sup>+1</sup>/(<i>n</i>+1)+<i>C</i><o:p></o:p></span></div>
</td>
  <td style="border-bottom: solid #CCCCCC 1.0pt; border-left: none; border-right: solid #CCCCCC 1.0pt; border-top: none; mso-border-alt: solid #CCCCCC .75pt; mso-border-left-alt: solid #CCCCCC .75pt; mso-border-top-alt: solid #CCCCCC .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt;"><div class="MsoNormal" style="margin-bottom: 3.75pt;">
<span style="font-family: inherit; font-size: 17pt; line-height: 115%;">∫<span class="apple-converted-space"><i> </i></span><i>x</i><sup>2</sup><i>dx</i><span class="apple-converted-space"> </span>=<span class="apple-converted-space"> </span><i>x</i><sup>2+1</sup>/(2+1)+<i>C</i><o:p></o:p></span></div>
</td>
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</tbody></table>
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<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>

Some Definations College Algebra

ALGEBRAIC EXPRESSION is an expression built up from constants , variables , and the algebraic operations ( addition , subtraction...

Fundamentals of Mathematics

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<span style="background-color: white; font-size: large;">Fractions</span></div>
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Frustrations with Fractions</span></h4>
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<span style="background-color: white;">If fractions get you frustrated, you are not alone. Many students are wary when it comes to working with fractions. We are here to help you get more comfortable!</span></div>
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<span style="background-color: white;">Simply put, a fraction is used to describe a part of a whole. The numerator (top) refers to the part and the denominator (bottom) describes the whole. In the fraction <sup>2</sup>⁄<sub>3</sub>, the numerator is 2 and the denominator is 3.</span></div>
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Equivalent Fractions</span></h4>
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<span style="background-color: white;">Two fractions that describe the same part of a whole are considered equivalent fractions. If two friends cut a pie in half, they can each take <sup>1</sup>⁄<sub>2</sub> of the whole. What if they both cut their piece of the pie into two pieces? Now, they each have two pieces of the pie, which is in four pieces. So, they each have <sup>2</sup>⁄<sub>4</sub> of the pie.<sup>2</sup>⁄<sub>4</sub> and <sup>1</sup>⁄<sub>2</sub> are equivalent fractions.</span></div>
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<span style="background-color: white;">Two equivalent fractions will be the same when completely reduced (written in simplest form). When one or both of the fractions contain large numbers, it may not be in your best interest to try to reduce or simplify them, although you certainly could do this. A more efficient way is to cross multiply. For example, say you are asked to compare the following two fractions:</span></div>
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<span style="background-color: white;"><img alt="fraction1728-(1).PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fraction1728-(1).PNG.aspx" style="height: 43px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 108px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">You can cross multiply as follows (denominator multiplied by the opposite numerator):</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="CrossMulti92-(1).PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/CrossMulti92-(1).PNG.aspx" style="height: 28px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 403px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Since the cross products are the same, the fractions are equivalent. If the product on the left were greater than the product on the right, the fraction on the left would be the larger of the two. Similarly, had the product on the right been larger, the fraction on the right would be the larger of the two.</span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">For example:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionsum1.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionsum1.PNG.aspx" style="height: 72px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 288px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Since the product on the left is smaller, the fraction on the left is also smaller.</span></div>
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<span style="background-color: white;">
Reducing Fractions</span></h4>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Reducing and simplifying fractions requires you to be able to identify the common factors between the numerator and the denominator. The divisibility rules come in handy here! You are looking for factors that can divide both the numerator and the denominator. One way to do this is to write out the prime factorization of both. Another is by inspection. Let’s use the following example:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fraction420.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fraction420.PNG.aspx" style="height: 45px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 38px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">If you wanted to write out the prime factorization and cancel out any common factors, you’d end up with the following:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum2.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum2.PNG.aspx" style="height: 55px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 184px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Alternatively, you can use inspection to look for common factors. Since both 420 and 630 end in zeros, they are both divisible by 10. This leaves you with <sup>42</sup>⁄<sub>63</sub>. Both 42 and 63 are divisible by 3 (the sums of the digits in each number are 6 and 9, respectively),leaving you with <sup>14</sup>⁄<sub>21.</sub> From here, it’s fairly easy to see that both are divisible by 7, leaving you with the most reduced form: <sup>2</sup>⁄<sub>3.</sub></span></div>
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<span style="background-color: white;">
Improper Fractions</span></h4>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Improper fractions typically make an appearance on the SAT. They are simply fractions that represent a whole number. You probably haven’t encountered them in math class since grade school however. When you see an improper fraction on the test, you will typically see answer choices with mixed numbers, meaning numbers that represent an improper fraction as a whole number and a fractional component. For example, <sup>7</sup>⁄<sub>3</sub> is an improper fraction. Three (3) divides into 7 twice with a remainder of 1. Thus, <sup>7</sup>⁄<sub>3</sub> could be written as the mixed number 2<sup>1</sup>⁄<sub>3</sub>.</span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Other times, you’ll need to do the same conversion, but in reverse, converting a mixed number into an improper fraction. This happens most often when multiplying or dividing with fractions. To do this, simply multiply the whole number by the denominator, add the result to the numerator, and divide the sum by the denominator. Let’s use our previous example, 2<sup>1</sup>⁄<sub>3</sub>. Writing 2<sup>1</sup>⁄<sub>3</sub> as an improper fraction would look something like:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionsum3.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionsum3.PNG.aspx" style="height: 43px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 117px;" /></span></div>
<h4 style="font-family: arial, sans-serif; font-size: 1.1em; letter-spacing: -1px; text-align: left;">
<span style="background-color: white;">
Common Denominators</span></h4>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">A common problem encountered when dealing with fractions involves finding a common denominator, or more importantly, the <em>least common denominator</em>. Before you can add or subtract fractions, you will need to find a common denominator.</span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">For example, let’s say you have the following problem:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionsum4.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionsum4.PNG.aspx" style="height: 46px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 68px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">The simplest way to find a common denominator is to take the product of the two denominators.</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum5.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum5.PNG.aspx" style="height: 24px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 74px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Using this as the common denominator, you would write the fractions, thus:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum6.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum6.PNG.aspx" style="height: 46px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 204px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">Then you can add them:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum7.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum7.PNG.aspx" style="height: 47px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 93px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">This can be done fairly quickly. However, it may result in answers requiring a significant amount of simplification as the common denominator found will not necessarily be the least common denominator. Finding this requires a couple additional steps on the front end. Let’s take the following problem:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum8.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum8.PNG.aspx" style="height: 51px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 98px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">First, we factor both the denominators:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum9.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum9.PNG.aspx" style="height: 48px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 257px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">The least common denominator will thus be:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionSum10.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionSum10.PNG.aspx" style="height: 26px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 193px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">All we need to do now is multiply the numerator and denominator of each fraction by the missing factors:</span></div>
<div align="center" style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px;">
<span style="background-color: white;"><img alt="fractionmissingeq.PNG" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionmissingeq.PNG.aspx" style="height: 46px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 319px;" /></span></div>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;">We are still dealing with fractions with rather large numbers, but this did make things simpler. Had we simply multiplied the two denominators together, we would have gotten a common denominator of 138,600!</span></div>
<h4 style="font-family: arial, sans-serif; font-size: 1.1em; letter-spacing: -1px; text-align: left;">
<span style="background-color: white;">
Example</span></h4>
<div style="font-family: Tahoma, Verdana, Arial; font-size: 12px; margin-bottom: 15px; margin-top: 8px; padding: 0px; text-align: left;">
<span style="background-color: white;"><img alt="Fractions Example" src="https://www.mycollegeoptions.org/Core/getattachment/SiteContent/Students/Tools/TestPrep/SAT/Math/Fractions/fractionExample-(1).PNG.aspx" style="height: 271px; margin: 0px 10px 0px 0px; padding-top: 2px; width: 314px;" /></span></div>
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<span style="background-color: white;">
Answer and Explanation</span></h4>
<ol class="solution" style="font-family: arial, sans-serif; font-size: 12px; line-height: 15px; text-align: left;">
<li style="color: rgb(34, 34, 34) !important; font-family: Tahoma, Verdana, Arial !important;"><span style="background-color: white;"><strong>The correct answer is A. </strong>This is a time-consuming problem, but nothing difficult. Check out your answer choices first and you’ll see that every option contains the option in I. Thus, there is no point in checking it, it must be greater! Start with the value in II. You can either use your calculator or the comparison method discussed above. The value in II is greater so you can eliminate choice E which doesn’t include it. Moving on to the value in III, it is also greater, so the correct response must contain III, eliminating choices C and D. You are left with choices A and B, so forget looking at IV, you only need to check the value in V which is also greater. Eliminate choice B.</span></li>
</ol>
</div>

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