# Negative Integer Exponents

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1 7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions that contain negative exponents In Section.4, all the exponents we looked at were positive integers. In this section, we look at the meaning of zero and negative integer exponents. First, let s look at an application of the quotient rule that will yield a zero exponent. Recall that, in the quotient rule, to divide expressions with the same base, keep the base and subtract the exponents. a m a n am n Now, suppose that we allow m to equal n. We then have a m () a m am m a 0 But we know that it is also true that a m (2) a m Comparing equations () and (2), we see that the following definition is reasonable. Rules and Properties: The Zero Exponent NOTE We must have a 0. The form 0 0 is called indeterminate and is considered in later mathematics classes. For any real number a when a 0, a 0 Example The Zero Exponent Use the above definition to simplify each expression. NOTE Notice that in 6x 0 the exponent 0 applies only to x. (a) 7 0 (b) (a 3 b 2 ) 0 (c) 6x (d) 3y McGraw-Hill Companies CHECK YOURSELF Simplify each expression. (a) 25 0 (b) (m 4 n 2 ) 0 (c) 8s 0 (d) 7t 0 Recall that, in the product rule, to multiply expressions with the same base, keep the base and add the exponents. a m a n a m+n 555

2 556 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS Now, what if we allow one of the exponents to be negative and apply the product rule? Suppose, for instance, that m 3 and n 3. Then a m a n a 3 a 3 a 3 ( 3) so a 0 a 3 a 3 Dividing both sides by a 3, we get a 3 a 3 Rules and Properties: Negative Integer Exponents NOTE John Wallis (66 702), an English mathematician, was the first to fully discuss the meaning of 0, negative, and rational exponents. For any nonzero real number a and whole number n, a n a n and a n is the multiplicative inverse of a n. Example 2 illustrates this definition. Example 2 Using Properties of Exponents NOTE From this point on, to simplify will mean to write the expression with positive exponents only. Simplify the following expressions. (a) y 5 y 5 NOTE Also, we will restrict all variables so that they represent nonzero real numbers. (b) (c) ( 3) 3 ( 3) (d) CHECK YOURSELF 2 (a) a 0 (b) 2 4 (c) ( 4) 2 (d) McGraw-Hill Companies

3 NEGATIVE INTEGER EXPONENTS SECTION Example 3 illustrates the case in which coefficients are involved in an expression with negative exponents. As will be clear, some caution must be used. Example 3 Using Properties of Exponents CAUTION (a) 2x 3 2 x 3 2 x 3 The expressions 4w 2 and (4w) 2 are not the same. Do you see why? The exponent 3 applies only to the variable x, and not to the coefficient 2. (b) 4w 2 4 w 2 4 w 2 (c) (4w) 2 (4w) 2 6w 2 CHECK YOURSELF 3 (a) 3w 4 (b) 0x 5 (c) (2y) 4 (d) 5t 2 Suppose that a variable with a negative exponent appears in the denominator of an expression. Our previous definition can be used to write a complex fraction that can then be simplified. For instance, a 2 a2 a2 a 2 Negative exponent in denominator. Positive exponent in numerator. To divide, we invert and multiply. To avoid the intermediate steps, we can write that, in general, 200 McGraw-Hill Companies Rules and Properties: For any nonzero real number a and integer n, a n a n Negative Exponents in a Denominator

4 558 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS Example 4 Using Properties of Exponents (a) (b) (c) y 3 y x 2 3x2 4 The exponent 2 applies only to x, not to 4. (d) a 3 b 4 b4 a 3 CHECK YOURSELF 4 2 (a) (b) (c) (d) c 5 x a 2 d 7 NOTE To review these properties, return to Section.4. The product and quotient rules for exponents apply to expressions that involve any integer exponent positive, negative, or 0. Example 5 illustrates this concept. Example 5 Using Properties of Exponents Simplify each of the following expressions, and write the result, using positive exponents only. (a) x 3 x 7 x 3 ( 7) (b) x 4 x 4 m 5 m 3 m 5 ( 3) m 5 3 Add the exponents by the product rule. Subtract the exponents by the quotient rule. m 2 m 2 NOTE Notice that m 5 in the numerator becomes m 5 in the denominator, and m 3 in the denominator becomes m 3 in the numerator. We then simplify as before. (c) x 5 x 3 x 7 x5 ( 3) x 7 x2 x 7 x2 ( 7) x 9 In simplifying expressions involving negative exponents, there are often alternate approaches. For instance, in Example 5(b), we could have made use of our earlier work to write m 5 m 3 m3 m 5 m 3 5 m 2 m 2 We apply first the product rule and then the quotient rule. 200 McGraw-Hill Companies

5 NEGATIVE INTEGER EXPONENTS SECTION CHECK YOURSELF 5 y 7 (a) (b) (c) a 3 a 2 x 9 x 5 y 3 a 5 The properties of exponents can be extended to include negative exponents. One of these properties, the quotient-power rule, is particularly useful when rational expressions are raised to a negative power. Let s look at the rule and apply it to negative exponents. Rules and Properties: a n a, b 0 b n b n Quotient-Power Rule Rules and Properties: a b n a n bn n b a n b a n Raising Quotients to a Negative Power a 0, b 0 Example 6 Extending the Properties of Exponents Simplify each expression. (a) s3 t 2 2 t 2 s 3 2 t4 s 6 (b) m 2 n 2 3 n 2 CHECK YOURSELF 6 Simplify each expression. s3 m 3 2 n 6 m 6 n 6 m 6 x (a) (b) 5 y 3 3t As you might expect, more complicated expressions require the use of more than one of the properties for simplification. Example 7 illustrates such cases. 200 McGraw-Hill Companies Example 7 Using Properties of Exponents (a) (a 2 ) 3 (a 3 ) 4 (a 3 ) 3 a 6 a 2 a 9 a 6 2 a 9 a6 a 9 a 6 ( 9) a 6 9 a 5 Apply the power rule to each factor. Apply the product rule. Apply the quotient rule.

6 560 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS NOTE It may help to separate the problem into three fractions, one for the coefficients and one for each of the variables. CAUTION (b) (c) 8x 2 y 5 2x 4 y x 2 x y 5 4 y x 2 ( 4) y x2 y 8 2x2 3y 8 pr3 s 5 p 3 r 3 s 2 2 p 3 r 3 s 2 pr 3 s 5 2 p6 r 6 s 4 p 2 r 6 s 0 p 4 r 2 s 6 p4 s 6 r 2 Be Careful! Another possible first step (and generally an efficient one) is to rewrite an expression by using our earlier definitions. a n a n and n an a For instance, in Example 8(b), we would correctly write 8x 2 y 5 2x 4 y 3 8x4 2x 2 y 3 y 5 A common error is to write 8x 2 y 5 2x 4 y 3 2x4 8x 2 y 3 y 5 This is not correct. The coefficients should not be moved along with the factors in x. Keep in mind that the negative exponents apply only to the variables. The coefficients remain where they were in the original expression when the expression is rewritten by using this approach. CHECK YOURSELF 7 (x 5 ) 2 (x 2 ) 3 2a 3 b 2 xy (a) (b) (c) 3 z 5 x 4 y 2 z 3 (x 4 ) 3 6a 2 b 3 3 CHECK YOURSELF ANSWERS 4. (a) ; (b) ; (c) 8; (d) 7 2. (a) ; (b) ; (c) ; (d) a a 2 d 7 3. (a) ; (b) ; (c) ; (d) 5 4. (a) x 4 ; (b) 27; (c) ; (d) w 4 x 5 6y 4 t 2 3 c 5 27t (a) x 4 ; (b) ; (c) a 4 6. (a) ; (b) 7. (a) x 8 ; (b) ; (c) y3 z 24 y 4 s 9 x 5 y 6 4ab 5 x McGraw-Hill Companies

7 Name 7.7 Exercises Section Date In exercises to 22, simplify each expression.. x ANSWERS x 8 5. ( 5) 2 6. ( 3) ( 2) 3 8. ( 2) x x x 4 4. ( 2x) 4 5. ( 3x) x x 3 x x 3 x 5 y 3 4x 4 x 3 y In exercises 23 to 32, use the properties of exponents to simplify the expressions. 23. x 5 x y 4 y a 9 a w 5 w z 2 z b 7 b McGraw-Hill Companies 29. a 5 a x 4 x x 3 x 6 x 5 x 2 56

8 ANSWERS In exercises 33 to 58, use the properties of exponents to simplify the following. 33. (x 5 ) (w 4 ) (2x 3 )(x 2 ) (p 4 )(3p 3 ) (3a 4 )(a 3 )(a 2 ) 38. (5y 2 )(2y)(y 5 ) (x 4 y)(x 2 ) 3 (y 3 ) (r 4 ) 2 (r 2 s)(s 3 ) 2 4. (ab 2 c)(a 4 ) 4 (b 2 ) 3 (c 3 ) (p 2 qr 2 )(p 2 )(q 3 ) 2 (r 2 ) (x 5 ) (x 2 ) (b 4 ) (a 0 b 4 ) (x 5 y 3 ) (p 3 q 2 ) (x 4 y 2 ) (3x 2 y 2 ) a 6 b 4 5. (2x 3 y 0 ) x 2 y x 3 x y 3 2 y (4x 2 ) 2 (3x 4 ) 58. (5x 4 ) 4 (2x 3 ) 5 (3x 4 ) 2 (2x 2 ) x 6 In exercises 59 to 90, simplify each expression. 59. (2x 5 ) 4 (x 3 ) (3x 2 ) 3 (x 2 ) 4 (x 2 ) 6. (2x 3 ) 3 (3x 3 ) (x 2 y 3 ) 4 (xy 3 ) (xy 5 z) 4 (xyz 2 ) 8 (x 6 yz) (x 2 y 2 z 2 ) 0 (xy 2 z) 2 (x 3 yz 2 ) 65. (3x 2 )(5x 2 ) (2a 3 ) 2 (a 0 ) (2w 3 ) 4 (3w 5 ) McGraw-Hill Companies 562

9 ANSWERS 3x 6 x (3x 3 ) 2 (2x 4 ) y 6 2y9 2y 9 y5 x 3 x 3 7. ( 7x 2 y)( 3x 5 y 6 ) w5 z (2x 2 y 3 )(3x 4 y 2 ) 3x 3 y x5 y 4 9 w 4 z (x 3 )(y 2 ) 74. ( 5a 2 b 4 )(2a 5 b 0 ) y 3 6x 3 y 4 24x 2 y x 3 y 2 z 4 24x 5 y 3 z x 4 y 3 z 2 36x 2 y 3 z x 2 y 2 8. x 3 y 2 x 4 y xy3 z 4 2 x 3 y 2 z 2 2 x 2 y x 5 y 7 x 0 y 4 x 3 y 3 x 4 y 3 x 2 2 y 2 xy x 2n x 3n 84. x n x 3n 85. x n 4 x n x n 3 x n (y n ) 3n 88. (x n ) n x 2n x n x 3n x n x 3n 5 x 4n Can (a b) be written as by using the properties of exponents? If not, why a b not? Explain Write a short description of the difference between ( 4) 3, 4 3, ( 4) 3, and 4 3. Are any of these equal? McGraw-Hill Companies 93. If n 0, which of the following expressions are negative? ( n) 3, n 3, n 3, ( n) 3, ( n) 3, n 3 If n 0, which of these expressions are negative? Explain what effect a negative in the exponent has on the sign of the result when an exponential expression is simplified

10 Answers x x 2 x x 3 2x 3 y x x 2 5 x 3 a 3 z x x x a 39. x 0 y 4. a 7 b 8 c b 8 x x 2 y 6 x 5 5. x 5 y 6 32 y 4 y x x 42 y 33 z 25 x x w 2 3x x 22 y y 4 x 2 y 5 y 5 3xy 5 y 8 5n x 85. x 2 x 3 4z 6 x 5 y 3 x x y 3n2 x 4 x 8 x McGraw-Hill Companies 564

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