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Laws of exponents notes

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    Laws of exponents notes Laws of exponents notes Document Transcript

    • OakRoadSystems → Articles → Math → Exponent Laws It’s the Law — the Laws of Exponents revised 6 Mar 2011 Copyright © 2002–2012 by Stan Brown, Oak Road SystemsSummary: The rules for combining powers and roots seem to confuse a lot of students. They try to memorize everything, and of course it’s a big mishmash in their minds. But the laws just come down to counting, which anyone can do, plus three de initions to memorize. This page sorts out what you have to memorize and what you can do based on counting, to solve every problem involving exponents.Contents: What Is an Exponent, Anyway? Negative Exponents Fractional Exponents Arbitrary Exponents Here’s All You Need to Memorize Now You Try It! Multiplying and Dividing Powers Two Powers of the Same Base Powers of Different Bases Dividing Powers Negative Powers on the Bottom Now You Try It! Powers of Powers Now You Try It! The Zero Exponent Now You Try It! Radicals Fractional or Rational Exponents Now You Try It! Conclusion Answers What’s NewSee also: Combining Operations (Distributive Laws) includes lots of common mistakes students make, with plenty of exercises to test yourself.Copying: You’re welcome to print copies of this page for your own use, and to link from your own Web pages to this page. But please don’t make any electronic copies and publish them on your Web page or elsewhere. 1 of 8
    • What Is an Exponent, Anyway?There’s nothing mysterious! An exponent is simply shorthand for multiplying that number ofidentical factors. So 4³ is the same as (4)(4)(4), three identical factors of 4. And x³ is just three factorsof x, (x)(x)(x). One warning: Remember the order of operations. Exponents are the irst operation (in the absenceof grouping symbols like parentheses), so the exponent applies only to what it’s directly attached to.3x³ is 3(x)(x)(x), not (3x)(3x)(3x). If we wanted (3x)(3x)(3x), we’d need to use grouping: (3x)³. Negative ExponentsA negative exponent means to divide by that number of factors instead of multiplying. So 4−3 is thesame as 1/(43), and x‐3 = 1/x3. As you know, you can’t divide by zero. So there’s a restriction that x−n = 1/xn only when x is notzero. When x = 0, x−n is unde ined. A little later, we’ll look at negative exponents in the bottom of a fraction. Fractional ExponentsA fractional exponent—speci ically, an exponent of the form 1/n—means to take the nth root insteadof multiplying or dividing. For example, 4(1/3) is the 3rd root (cube root) of 4. Arbitrary ExponentsYou can’t use counting techniques on an expression like 60.1687 or 4.3π. Instead, these expressions areevaluated using logarithms. Here’s All You Need to MemorizeAnd that’s it for memory work. Period. If you memorize these three de initions, you can workeverything else out by combining them and by counting: 2 of 8
    • Granted, there’s a little bit of hand waving in my statement that you can work everything else out. Letme make good on that promise, by showing you how all the other laws of exponents come from just thethree de initions above. The idea is that you won’t need to memorize the other laws—or if you dochoose to memorize them, you’ll know why they work and you’ll ind them easier to memorizeaccurately. Now You Try It!1. Write 11³ as a multiplication.2. Write j‐7 as a fraction, using only positive exponents.3. What’s the value of 100½?4. Evaluate −5‐2 and (−5)‐2. [ Answers ]Multiplying and Dividing Powers Two Powers of the Same BaseSuppose you have (x5)(x6); how do you simplify that? Just remember that you’re counting factors. x5 = (x)(x)(x)(x)(x) and x6 = (x)(x)(x)(x)(x)(x)Now multiply them together: (x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11Why x11? Well, how many x’s are there? Five x factors from x5, and six x factors from x6, makes 11 xfactors total. Can you see that whenever you multiply any two powers of the same base, you end upwith a number of factors equal to the total of the two powers? In other words, when the bases are thesame, you ind the new power by just adding the exponents: 3 of 8
    • Powers of Different BasesCaution! The rule above works only when multiplying powers of the same base. For instance, (x3)(y4) = (x)(x)(x)(y)(y)(y)(y)If you write out the powers, you see there’s no way you can combine them. Except in one case: If the bases are different but the exponents are the same, then you cancombine them. Example: (x³)(y³) = (x)(x)(x)(y)(y)(y)But you know that it doesn’t matter what order you do your multiplications in or how you group them.Therefore, (x)(x)(x)(y)(y)(y) = (x)(y)(x)(y)(x)(y) = (xy)(xy)(xy)But from the very de inition of powers, you know that’s the same as (xy)³. And it works for any commonpower of two different bases:It should go without saying, but I’ll say it anyway: all the laws of exponents work in both directions. Ifyou see (4x)³ you can decompose it to (4³)(x³), and if you see (4³)(x³) you can combine it as (4x)³. Dividing PowersWhat about dividing? Remember that dividing is just multiplying by 1‐over‐something. So all the lawsof division are really just laws of multiplication. The extra de inition of x‐n as 1/xn comes into play here. Example: What is x8÷x6? Well, there are several ways to work it out. One way is to say that x8÷x6 =x 8(1/x6), but using the de inition of negative exponents that’s just x8(x‐6). Now use the product rule(two powers of the same base) to rewrite it as x8+(‐6), or x8‐6, or x2. Another method is simply to goback to the de inition: x8÷x6 = (xxxxxxxx)÷(xxxxxx) = (xx)(xxxxxx)÷(xxxxxx) = (xx)(xxxxxx÷xxxxxx) =(xx)(1) = x2. However you slice it, you come to the same answer: for division with like bases yousubtract exponents, just as for multiplication of like bases you add exponents:But there’s no need to memorize a special rule for division: you can always work it out from the otherrules or by counting.In the same way, dividing different bases can’t be simpli ied unless the exponents are equal. x³÷y²can’t be combined because it’s just xxx/yy; But x³÷y³ is xxx/yyy, which is (x/y)(x/y)(x/y), which is(x/y)³. 4 of 8
    • Negative Powers on the BottomWhat about dividing by a negative power, like y5/x−4? Use the rule you already know for dividing: 5 5 5 4 5 4 y y y x y x 4 5 --- = -------- = -------- * -- = ----- = x y -4 ( 4) ( 4) 4 x (1 / x ) (1 / x ) x 1But that’s much too elaborate. Since 1 / (1/x) is just x, a negative exponent just moves its power to theother side of the fraction bar. So x−4 = 1/(x4), and 1/(x−4) = x4. Now You Try It!Write each of these as a single positive power. (I’ve slipped in one or two that can’t be simpli ied, just tokeep you on your toes.) 5. a7 ÷ b7 6. 11² × 2³ 7. 8³ x³ 8. 54 × 56 9. p11 ÷ p6 10. r‐11 ÷ r‐2 [ Answers ]Powers of PowersWhat do you do with an expression like (x5)4? There’s no need to guess—work it out by counting. (x5)4 = (x5)(x5)(x5)(x5)Write this as an array: x5 = (x) (x) (x) (x) (x) x5 = (x) (x) (x) (x) (x) x5 = (x) (x) (x) (x) (x) x5 = (x) (x) (x) (x) (x)How many factors of x are there? You see that there are 5 factors in each row from x5 and 4 rows from( )4, in all 5×4=20 factors. Therefore, (x5)4 = x20 5 of 8
    • As you might expect, this applies to any power of a power: you multiply the exponents. For instance,(k‐3)‐2 = k(‐3)(‐2) = k6. In general,I can just hear you asking, “So when do I add exponents and when do I multiply exponents?” Don’t try toremember a rule—work it out! When you have a power of a power, you’ll always have a rectangulararray of factors, like the example above. Remember the old rule of length×width, so the combinedexponent is formed by multiplying. On the other hand, when you’re only multiplying two powerstogether, like g2g3, that’s just the same as stringing factors together, g2g3 = (gg)(ggg) = (ggggg) = g5You can always refresh your memory by counting simple cases, like x2x3 = (xx)(xxx) = x5versus (x2)3 = (xx)3 = (xx)(xx)(xx) = x6 Now You Try It!Perform the operations to remove parentheses: 11. (x4)‐5 12. (5x²)³ [ Answers ]The Zero ExponentYou probably know that anything to the 0 power is 1. But now you can see why. Consider x0. By the division rule, you know that x3/x3 = x(3−3) = x0. But anything divided by itself is 1, sox3/x3 = 1. Things that are equal to the same thing are equal to each other: if x3/x3 is equal to both 1 andx0, then 1 must equal x0. Symbolically, x0 = x(3−3) = x3/x3 = 1There’s one restriction. You saw that we had to create a fraction to igure out x0. But division by 0 is notallowed, so our evaluation works for anything to the 0 power except zero itself:Evaluating 00 is a topic for your calculus course. Now You Try It! 6 of 8
    • What is the value of each of these? 13. (a6b8c10 / a5b6d7)0 14. 17x0 [ Answers ]RadicalsThe laws of radicals are traditionally taught separately from the laws of exponents, and frankly I’venever understood why. A radical is simply a fractional exponent: the square (2nd) root of x is justx1/2, the cube (3rd) root is just x1/3, and so on. With this fact at your disposal, you’re in good shape. Example: . That’s easy to evaluate! You know that the cube (3rd) root of x is x1/3 and thesquare root of that is (x1/3)1/2. Then use the power‐of‐a‐power rule to evaluate that as x(1/3)(1/2) =x(1/6), which is the 6th root of x. Example: . Why? Because the square root is the 1/2 power, and the productrule for the same power of different bases tells you that (x1/2)(y1/2) = (xy)1/2. Fractional or Rational ExponentsSo far we’ve looked at fractional exponents only where the top number was 1. How do you interpretx2/3, for instance? Can you see how to use the power rule? Since 2/3 = (2)(1/3), you can rewrite x2/3 =x(2)(1/3) = (x2)1/3, which is . It works the other way, too: 2/3 = (1/3)(2), so x2/3 = x(1/3)(2) =(x1/3)2 = . These are examples of the general rule:When a power and a root are involved, the top part of the fractional exponent is the power and thebottom part is the root. Suppose p and r are the same? Then you have, for instance, . But that’s the same as x5/5, and5/5=1, so it’s the same as x1 or just x. Now You Try It!15. Write √x5 as a single power.16. Simplify ³√(a6b9) (That’s the cube root or third root of a6b9.)17. Find the numerical value of 274/3 without using a calculator. [ Answers ] 7 of 8
    • ConclusionWell, there you are: the laws of exponents and radicals demysti ied! Just remember the three basicde initions. When you’re not sure about a rule, like the product rule, don’t try to remember it, just workit out by counting and you’ll do just ine.AnswersFrom What Is an Exponent, Anyway? — 1. 11×11×11 2. 1/j7 3. √100 = 10 4. −5‐2 = −1/25 and(−5)‐2 = 1/25 (Excel returns 1/25 or 0.04 for both of these, but that’s wrong.)From Multiplying and Dividing Powers — 5. (a/b)7 6. cannot be simpli ied. 7. (8x)³ 8. 510(not 2510!) 9. p5 10. r‐11‐(‐2) = r‐9 = 1/r9From Powers of Powers — 11. x‐20 or 1/x20 12. Use the rule for powers of different bases to startwith: 53(x2)3. Then apply the power‐of‐a‐power rule to get 53x6 or 125x6From The Zero Exponent — 13. 1 14. 17×1 = 17From Radicals — 15. x5/2 16. (a6b9)1/3 = a²b³ 17. 274/3 = (271/3)4 by the power‐of‐a‐power law.271/3 is the same as the cube root of 27, which is 3. (271/3)4 = 34 = 81What’s New 6 Mar 2011: Add an example with negative powers of negative bases, as suggested by Joshua Sararas; add Arbitrary Exponents 28 Feb 2010: change 11²×4³ to 11²×2³. (I had said the old example can’t be simpli ied to 445, but De i Abrasheva correctly pointed out that 4³=8² so the old example could be simpli ied to 88².) (intervening changes suppressed) 24 Feb 2002: new articlethis page: http://oakroadsystems.com/math/expolaws.htm 8 of 8