{"id":1917,"date":"2012-06-07T09:00:51","date_gmt":"2012-06-07T16:00:51","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=1917"},"modified":"2012-06-07T14:35:41","modified_gmt":"2012-06-07T21:35:41","slug":"exponent-properties-on-the-gmat","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/exponent-properties-on-the-gmat\/","title":{"rendered":"Exponent Properties on the GMAT"},"content":{"rendered":"

Because of their clarity and concision, the laws of exponents lend themselves well to GMAT math, especially to the Data Sufficiency format.\u00a0 If math isn’t your thing, then perhaps the last time you gave any thought to exponents was back in Algebra Two, and perhaps exponents weren’t your favorites there either.\u00a0 Take heart!\u00a0 In this post, I will explain the properties you need to know to be successful on the GMAT Quantitative section.<\/p>\n

 <\/p>\n

What is an Exponent?<\/h2>\n

Fundamentally, an exponent is how many times you multiply a number, that is, how many factors of a number you have.\u00a0 This is the fundamental definition<\/span>.\u00a0 The expression \"5^4\" means: multiply four 5’s together.\u00a0 The expression \"2^3\" means: multiply three 2’s together, which gives an answer of 8, so \"2^3.\u00a0 Technically, 2 is the “base”, 3 is the “exponent”, and 8 is the “power.”\u00a0 The action of raising something to an exponent is called “exponentiation.”<\/p>\n

 <\/p>\n

Distribution<\/h2>\n

Just as multiplication distributes over addition & subtraction<\/p>\n

\"a*(b<\/p>\n

so exponentiation distributes over multiplication and division.<\/p>\n

\"\"<\/p>\n

\"\"
\n<\/a>Why is that?\u00a0 Well, consider \"(x*y)^3\".\u00a0 This means the thing in parentheses multiplied by itself three times: \"(x*y)^3\u00a0.\u00a0 Well, when we have a bunch of factors, we can rearrange them in any order, because order doesn’t matter in multiplication.\u00a0 So, I could rearrange them as follows:<\/p>\n

\"(x*y)^3=\"<\/p>\n

\"(x*y)*(x*y)*(x*y)=\"<\/p>\n

\"x*x*x*y*y*y=\"<\/p>\n

\"(x*x*x)*(y*y*y)=\"<\/p>\n

\"(x^3)*(y^3)\"<\/p>\n

All the laws of exponents make sense if you just go back to the fundamental definition.<\/p>\n

In this context, I will say: beware of one of the most tempting mistakes in all of mathematics.\u00a0 Exponentiation does NOT distribute over addition.<\/p>\n

\"\"<\/p>\n

 <\/p>\n

 <\/p>\n

Beware.\u00a0 Even when you know this is wrong, even when you make an effort to remember that it’s wrong, the inherent pattern-matching machinery of your brain will automatically pull your mind back in the direction of making this mistake.\u00a0 You must be vigilant to avoid this mistake.<\/p>\n

 <\/p>\n

Multiplying Powers<\/h2>\n

What happens when you multiply two unequal powers of the same base?<\/p>\n

\"\"<\/a><\/p>\n

 <\/p>\n

 <\/p>\n

Well, let’s think about a concrete example.\u00a0 Suppose we are multiplying \"(x^5)*(x^3)\".\u00a0 Well, by the fundamental definition, \"x^5, and \"x^3, so<\/p>\n

\"(x^5)*(x^3)<\/p>\n

If we start with five factors of x, and “stir in” three more factors of x, we wind up with a total of eight factors.\u00a0 All we have to do is add the exponents.\u00a0 We can simply generalize this pattern:<\/p>\n

\"\"<\/a><\/p>\n

Don’t just memorize this: make sure you understand the logic that leads to it.\u00a0 Remembering with the logic is 100x more effective than blind memorization!<\/p>\n

 <\/p>\n

Dividing Powers<\/h2>\n

What happens when you divide two unequal powers of the same base?<\/p>\n

\"\"<\/a><\/p>\n

As with last time, a concrete example will illuminate the question.\u00a0 Suppose we divide \"(x^7)\/(x^3)\".\u00a0 By the fundamental definition, \"x^7 and \"x^3, so<\/p>\n

\"\"<\/a><\/p>\n

If we start out with seven factors, and then cancel away three of them, we are left with four.\u00a0 We just subtract the exponents.\u00a0 We can also generalize this pattern:<\/p>\n

\"\"<\/a><\/p>\n

Once again: understand the logic, because remember through understanding is considerably more powerful than blind memorization.<\/p>\n

 <\/p>\n

An Exponent of Zero<\/h2>\n

Mathematicians love to extend patterns.\u00a0 One example of this is the zero exponent.\u00a0 If we just see x^0, we may wonder: how on earth are we going to understand what this could mean?\u00a0 We are clearly outside of the realm where the fundamental definition helps us.<\/p>\n

One clever trick we can us is to employ the pattern found in division of powers.\u00a0 Suppose we have \"(x^4)\/(x^4)\" — then, the “subtraction of exponents” would imply:<\/p>\n

\"\"<\/a><\/p>\n

but just fundamental logic would tell us that anything over itself equals one.\u00a0 Therefore, this expression \"(x^4)\/(x^4)\" must have a value of 1.\u00a0 That, in turn, tells us the value of \"x^0\". \u00a0\"x^0.<\/p>\n

<\/h2>\n

Negative Exponents<\/h2>\n

Here, we will extend the patterns even further.\u00a0 Consider this chart, for a base of 2:<\/p>\n\n\n\n\n
Exponent<\/strong><\/td>\n\n

0<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

3<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n<\/tr>\n

Power<\/strong><\/td>\n\n

1<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n

\n

8<\/p>\n<\/td>\n

\n

16<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Each time we move one cell to the right, the power gets multiplied by 2, and each time we move one cell to the left, the power gets divided by 2.\u00a0 That’s a very easy pattern to extend to the left:<\/p>\n\n\n\n\n
Exponent<\/strong><\/td>\n\n

\u20134<\/p>\n<\/td>\n

\n

\u20133<\/p>\n<\/td>\n

\n

\u20132<\/p>\n<\/td>\n

\n

\u20131<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

3<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n<\/tr>\n

Power<\/strong><\/td>\n\n

1\/16<\/p>\n<\/td>\n

\n

1\/8<\/p>\n<\/td>\n

\n

1\/4<\/p>\n<\/td>\n

\n

1\/2<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n

\n

8<\/p>\n<\/td>\n

\n

16<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

All we have done was to extend the pattern “move one cell to the left, and the power gets divided by 2.”\u00a0 The result, we see, is that negative powers are reciprocals of their corresponding positive powers.\u00a0 This is consistent with the Division of Powers rule: if dividing means subtract the exponents, then an exponent of \u20133 means we are dividing by three factors of the number.\u00a0 Therefore, the general rule is:<\/p>\n

\"\"<\/a><\/p>\n

 <\/p>\n

Summary<\/h2>\n

I can’t urge enough: the key to remembering these rules is understanding the logic of the arguments behind them.\u00a0 If you understand these rules, you will understand whatever the GMAT throws at you concerning exponents.<\/p>\n

 <\/p>\n

Here are some practice questions:<\/p>\n

1) http:\/\/gmat.magoosh.com\/questions\/714<\/a><\/p>\n

2) http:\/\/gmat.magoosh.com\/questions\/715<\/a><\/p>\n

3) http:\/\/gmat.magoosh.com\/questions\/322<\/a><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Because of their clarity and concision, the laws of exponents lend themselves well to GMAT math, especially to the Data Sufficiency format.\u00a0 If math isn’t your thing, then perhaps the last time you gave any thought to exponents was back in Algebra Two, and perhaps exponents weren’t your favorites there either.\u00a0 Take heart!\u00a0 In this […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[150],"tags":[],"ppma_author":[13209],"class_list":["post-1917","post","type-post","status-publish","format-standard","hentry","category-basics"],"acf":[],"yoast_head":"\nExponent Properties on the GMAT | Magoosh Study Resources<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/exponent-properties-on-the-gmat\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Exponent Properties on the GMAT\" \/>\n<meta property=\"og:description\" content=\"Because of their clarity and concision, the laws of exponents lend themselves well to GMAT math, especially to the Data Sufficiency format.\u00a0 If math isn’t your thing, then perhaps the last time you gave any thought to exponents was back in Algebra Two, and perhaps exponents weren’t your favorites there either.\u00a0 Take heart!\u00a0 In this […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/exponent-properties-on-the-gmat\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-06-07T16:00:51+00:00\" \/>\n<meta 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M\u1d9cGarry","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/15a1e36ef1c2c3940179212433de141a","url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","caption":"Mike M\u1d9cGarry"},"description":"Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/1917","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=1917"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/1917\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=1917"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=1917"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=1917"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=1917"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}