Using Rational Exponents and nth Roots
Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2015 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: July 6, 2015
www.ck12.org
C HAPTER
Chapter 1. Using Rational Exponents and nth Roots
1
Using Rational Exponents and nth Roots
Objective To introduce rational exponents and nth roots. Then, we will apply the properties of exponents to rational functions and nth roots. Review Queue Evaluate each expression. 1. (5x)2 4x5 y7 2. 12xy 9 p 3. r81x2 y5 75 4. 96
Defining nth Roots Objective To define and use nth roots. Guidance So far, we have seen exponents with integers and the square root. In this concept, we will link roots and exponents. First, let’s define additional roots. Just like the square and the square root are inverses of each other, the inverse of a cube is the cubed root. The inverse of the fourth power is the fourth root. √ √ √ √ 5 3 3 5 27 = 33 = 3, 32 = 25 = 2 The nth root of a number, xn , is x,
√ n n x = x. And, just like simplifying square roots, we can simplify nth roots.
Example A √ 6 Find 729. Solution: To simplify a number to the sixth root, there must be 6 of the same factor to pull out of the root. 729 = 3 · 3 · 3 · 3 · 3 · 3 = 36 √ √ 6 6 Therefore, 729 = 36 = 3. The sixth root and the sixth power cancel each other out. We say that 3 is the sixth root of 729. From this example, we can see that it does not matter where the exponent is placed, it will always cancel out with the root. √ 6 √ √ 6 6 6 6 36 = 3 or 3 √ 6 729 = (1.2009 . . .)6 3=3 1
www.ck12.org So, it does not matter if you evaluate the root first or the exponent. The nth Root √ √ mTheorem: For any real number a, root n, and exponent m, the following is always true: m n a = na .
√ n am =
Example B Evaluate without a calculator: √ 5 a) 323 √ 3 b) 16 Solution: a) If you solve this problem as written, you would first find 323 and then apply the 5th root. √ √ 5 5 323 = 38768 = 8 However, this would be very difficult to do without a calculator. This is an example where it would be easier to apply the root and then the exponent. Let’s rewrite the expression and solve. √ 3 5 32 = 23 = 8 √ b) This problem does not need to be rewritten. 16 = 4 and then 43 = 64. Example C Simplify: √ 4 a) 64 s 54x3 3 b) 125y5 Solution: a) To simplify the fourth root of a number, there must be 4 of the same factor to pull it out of the root. Let’s write the prime factorization of 64 and simplify. √ √ √ 4 4 4 64 = 2 · 2 · 2 · 2 · 2 · 2 = 2 4 Notice that there are 6 2’s in 64. We can pull out 4 of them and 2 2’s are left under the radical. b) Just like simplifying fractions with square roots, we can separate the numerator and denominator. s √ √ √ 3 3 3 3 3 54x3 3 3xp 2 54x 2 · 3 · 3 · 3 · x p p = = = 3 3 125y5 5 · 5 · 5 · y3 · y2 5y 3 y2 125y5 Notice that because the x is cubed, the cube and cubed root cancel each other out. With the y-term, there were five, so three cancel out with the root, but two are still left under radical. Guided Practice Simplify each expression below, without a calculator. p 4 1. 625z8 p 7 2. 32x5 y √ 5 3. 9216 r 40 3 4. 175 Answers 1. First, you can separate this number into two different roots, 2
p √ 4 4 625 · z8 . Now, simplify each root.
www.ck12.org
Chapter 1. Using Rational Exponents and nth Roots
p p √ √ 4 4 4 4 625 · z8 = 54 · z4 · z4 = 5z2 When looking at the z8 , think about how many z4 you can even pull out of the fourth root. The answer is 2, or a z2 , outside of the radical. 2. 32 = 25 , which means there are not 7 2’s that can be pulled out of the radical. Same with the x5 and the y. Therefore, you cannot simplify the expression any further. 3. Write out 9216 in the prime factorization and place factors into groups of 5.
q √ 5 5 9216 = 2·2·2·2·2 · 2·2·2·2·2 ·3·3 √ 5 = 25 · 25 · 32 √ 5 = 2 · 2 32 √ 5 =4 9 4. Reduce the fraction, separate the numerator and denominator and simplify. r r √ √ √ 3 3 3 40 8 3 3 23 = √2 · √352 = 2 1225 = = √ 3 3 3 35 175 35 35 35 352
√ 3 In the red step, we rationalized the denominator by multiplying the top and bottom by 352 , so that the denominator √ 3 would be 353 or just 35. Be careful when rationalizing the denominator with higher roots!
Vocabulary nth root √ n The nth root of a number, xn , is x, xn = x. Problem Set Reduce the following radical expressions. √ 3 1. √81 5 2. r128 4 25 3. 8 √ 5 6 4. r64 2 8 3 5. r 81 4 243 6. 16 √ 3 7. p24x5 4 8. s48x7 y13 9.
5
160x8 y7
√ 2 3 10. s1000x6 5 4 162x 11. y3 z10 p 3 12. 40x3 y4 3
www.ck12.org
Rational Exponents and Roots Objective To introduce rational exponents and relate them to nth roots. Guidance Now that you are familiar with nth roots, we will convert them into exponents. Let’s look at the square root and see if we can use the properties of exponents to determine what exponential number it is equivalent to. Investigation: Writing the Square Root as an Exponent
TABLE 1.1: 1. Evaluate
√ 2 x . What happens?
2. Recall that when a power is raised to another power, we multiply the exponents. Therefore, we can rewrite the exponents and root as an equation, n · 2 = 1. Solve for n. √ 3. From #2, we can conclude that = 12 . From this we see that √ investigation, 1 1 x4 ,... n x = xn .
√ and the 2 cancel each other out, The n·2 =1 2 2 n = 12
√ 2 x = x.
√ 2 1 2 ( 1 )·2 1 x = x2 = x 2 = x = x
√ √ √ 1 1 x = x 2 . We can extend this idea to the other roots as well; 3 x = x 3 = 4 x =
Example A 1
Find 256 4 . Solution: Rewrite this expression in terms of roots. A number to the one-fourth power is the same as the fourth root. √ √ 4 1 4 256 4 = 256 = 44 = 4 1
Therefore, 256 4 = 4. Example B 3
Find 49 2 . Solution: This problem is the same as the ones in the previous concept. However, now, the root is written in the exponent. Rewrite the problem. √ 3 √ 1 3 49 2 = 493 2 = 493 or 49 √ 3 From the previous concept, we know that it is easier to evaluate the second option above. 49 = 73 = 343. The Rational Exponent Theorem: For any real number a, root n, and exponent m, the following is always true: √ √ m m n a n = am = n a . Example C 2
Find 5 3 using a calculator. Round your answer to the nearest hundredth. 2
Solution: To type this into a calculator, the keystrokes would probably look like: 5 3 . The “^” symbol is used to indicate a power. Anything in parenthesis after the “^” would be in the exponent. Evaluating this, we have 2.924017738..., or just 2.92. Other calculators might have a xy button. This button has the same purpose as the ^ and would be used in the exact 4
www.ck12.org
Chapter 1. Using Rational Exponents and nth Roots
same way. Guided Practice √ 7 1. Rewrite 12 using rational exponents. Then, use a calculator to find the answer. 4
2. Rewrite 845 9 using roots. Then, use a calculator to find the answer. Evaluate without a calculator. 4
3. 125 3 5
4. 256 8 q 1 5. 81 2 Answers 1. Using rational exponents, the 7th root becomes the
1 7
1
power;12 7 = 1.426. √ 9 2. Using roots, the 9 in the denominator of the exponent is the root; 8454 = 19.99. To enter this into a calculator, √ you can use the rational exponents. If you have a TI-83 or 84, press MATH and select 5: x . On the screen, you √ should type 9 x 845∧ 4 to get the correct answer. You can also enter 845∧ 94 and get the exact same answer √ 4 4 3 3. 125 3 = 125 = 54 = 625 √ 5 5 8 256 = 25 = 32 4. 256 8 = q q √ √ 1 81 = 9 = 3 5. 81 2 = Vocabulary Rational Exponent An exponent that can be written as√a fraction. For any nth root, the n of the root can be written in the 1 denominator of a rational exponent. n x = x n . Problem Set Write the following expressions using rational exponents and then evaluate using a calculator. Answers should be rounded to the nearest hundredth. √ 5 1. √45 9 2. √140 3 8 3. 50 Write the following expressions using roots and then evaluate using a calculator. Answers should be rounded to the nearest hundredth. 5
4. 72 3 2 5. 95 3 3 6. 125 4 Evaluate the following without a calculator. 2
7. 64 3 4 8. 27 3 5 9. 16 4 5
www.ck12.org √ 10. √ 253 5 2 11. √9 5 12. 322 For the following problems, rewrite the expressions with rational exponents and then simplify the exponent and evaluate without a calculator. s 2 8 4 13. 3 r 6 3 7 14. q2
6
1
(16) 2
15.
Applying the Laws of Exponents to Rational Exponents Objective To use the laws of exponents with rational exponents. Guidance When simplifying expressions with rational exponents, all the laws of exponents that were learned in the Polynomial Functions chapter are still valid. On top of that, all the rules of fractions still apply. Example A 3
1
Simplify x 2 · x 4 . Solution: Recall from the Product Property of Exponents, that when two numbers with the same base are multiplied we add the exponents. Here, the exponents do not have the same base, so we need to find a common denominator and then add the numerators. 1
3
2
3
5
x2 ·x4 = x4 ·x4 = x4 This rational exponent does not reduce, so we are done. Example B 2
Simplify
4x 3 y4 5
16x3 y 6
Solution: This problem utilizes the Quotient Property of Exponents. Subtract the exponents with the same base and 4 reduce 16 . 2
4x 3 y4 5 16x3 y 6
2
= 41 x( 3 )−3 y
4−5 6
−7
19
= 14 x 3 y 6
If you are writing your answer in terms of positive exponents, your answer would be rational exponent is improper we do not change it to a mixed number.
19 6 7 4x 3
y
. Notice, that when a
If we were to write the answer using roots, then we would take out the whole numbers. For example, y = √ 19 1 written as y 6 = y3 y 6 = y3 6 y because 6 goes into 19, 3 times with a remainder of 1. Example C 6 52 Simplify 9x 10 . 6
19 6
can be
www.ck12.org
Chapter 1. Using Rational Exponents and nth Roots
Solution: This example uses the Powers Property of Exponents. When a power is raised to another power, we multiply the exponents. 6 52 √ 5 30 6 5 5 3 9x 15 = 9 2 · x( 10 )·( 2 ) = 9 x 20 = 243x 2 Example D 2 3 1 2x 2 y6
Simplify
5
9
.
4x 4 y 4
Solution: On the numerator, the entire expression is raised to the 23 power. Distribute this power to everything inside the parenthesis. Then, use the Powers Property of Exponents and rewrite 4 as 22 . 2 3 1 2x 2 y6
2
=
5 9 4x 4 y 4
1
2 3 x 3 y4 5 9 22 x 4 y 4
Combine like terms by subtracting the exponents. 2
1
2 3 x 3 y4 5 9 22 x 4 y 4
1 2 5 9 −4 −11 7 = 2( 3 )−2 x( 3 )−( 4 ) y4−( 4 ) = 2 3 x 12 y 4
Finally, rewrite the answer with positive exponents by moving the 2 and x into the denominator.
7 y4 4 11 2 3 x 12
Guided Practice Simplify each expression. Reduce all rational exponents and write final answers using positive exponents. 1
3
2
1. 4d 5 · 8 3 d 5 7
2.
w4 1 w2
3 34 6 4 5 2 3. 3 x y Answers 1. Change 4 and 8 so that they are powers of 2 and then add exponents with the same base. 1 2 1 2 3 5 3 4d 5 · 8 3 d 5 = 22 d 5 · 23 3 d 5 = 23 d 5 = 8d 2. Subtract the exponents. Change the 7
7
w4 1 w2
=
w4 2 w4
1 2
power to 42 .
5
= w4
3. Distribute the 43 power to everything inside the parenthesis and reduce. 3 34 6 12 16 24 16 8 16 8 3 2 x4 y 5 = 3 6 x 3 y 15 = 32 x 3 y 5 = 9x 3 y 5 Problem Set Simplify each expression. Reduce all rational exponents and write final answer using positive exponents.
4
3
3
1. 15 a 5 25 2 a 5 4 2 1 2. 7b8 3 49 2 b− 3 3. m 92 4.
m3 4 11 x7 y 6 1
5
x 14 y 3
5.
5 3 1 8 3 r5 s 4 t 3 21 7 24 r 5 s2 t 9
7
www.ck12.org 3 4 103 6. a 2 b 5 5 32 7. 5x 7 y4 2 52 8. 4x 45 9y 5 18 52 9. 75d 35 3d 5 3 13 3 10. 81 2 9a 8a 2
2 3
4
3
1
2
8
− 2 4 2 m− 3 n 5 11. 27 m3 5 n2 2 3x 8 y 5 12. 1 − 3 5x 4 y
8
10