Copyright © Cengage Learning. All rights reserved. 9 Radicals and Rational Exponents Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. Section Rational Exponents 9.3 Copyright © Cengage Learning. All rights reserved.

Objectives Simplify an expression that contains a positive rational exponent with a numerator of 1. Simplify an expression that contains a positive rational exponent with a numerator other than 1. Simplify an expression that contains a negative rational exponent. 1 2 3

Objectives Simplify an expression that contains rational exponents by applying the properties of exponents. Simplify a radical expression by first writing it as an expression with a rational exponent. 4 5

Simplify an expression that contains a positive rational exponent with a numerator of 1 1.

Simplify an expression that contains a positive rational exponent with a numerator of 1 We have seen that positive integer exponents indicate the number of times that a base is to be used as a factor in a product. For example, x5 means that x is to be used as a factor five times. x5 = x  x  x  x  x 5 factors of x

Simplify an expression that contains a positive rational exponent with a numerator of 1 Furthermore, we recall the following properties of exponents. Rules of Exponents If there are no divisions by 0, then for all integers m and n, 1. xmxn = xm + n 2. (xm)n = xmn 3. (xy)n = xnyn 4. 5. x0 = 1 (x  0) 6. 7. 8. 9.

Simplify an expression that contains a positive rational exponent with a numerator of 1 To show how to raise bases to rational powers, we consider the expression 101/2. Since rational exponents must obey the same rules as integer exponents, the square of 101/2 is equal to 10. (101/2)2 = 10(1/2)2 = 101 = 10 However, we have seen that Keep the base and multiply the exponents. 101 = 10

Simplify an expression that contains a positive rational exponent with a numerator of 1 Since (101/2)2 and both equal to 10, we define 101/2 to be Likewise, we define 101/3 to be and 101/4 to be Rational Exponents If n is a natural number greater than 1, and is a real number, then

Example 1 Simplify each expression. Assume all variables represent nonnegative values.

Example 1 – Solution

Example 1 – Solution cont’d

Simplify an expression that contains a positive rational exponent with a numerator of 1 As with radicals, when n is even in the expression x1/n(n > 1), there are two real nth roots and we must use absolute value symbols to guarantee that the simplified result is positive. When n is odd, there is only one real nth root, and we do not need to use absolute value symbols. When n is even and x is negative, the expression x1/n is not a real number.

Simplify an expression that contains a positive rational exponent with a numerator of 1 We summarize the cases as follows. Summary of the Definitions of x1/n Assume n is a natural number greater than 1 and x is a real number. If x > 0, then x1/n is the positive number such that (x1/n)n = x. If x = 0, then x1/n = 0. and n is odd, then x1/n is the real number such that (x1/n)n = x. and n is even, then x1/n is not a real number. If x < 0,

Simplify an expression that contains a Simplify an expression that contains a positive rational exponent with a numerator other than 1 2.

Simplify an expression that contains a positive rational exponent with a numerator other than 1 We can extend the definition of x1/n to include rational exponents with numerators other than 1. For example, since 43/2 can be written as (41/2)3, we have 43/2 = (41/2)3 = = 23 = 8 Thus, we can simplify 43/2 by cubing the square root of 4. We can also simplify 43/2 by taking the square root of 4 cubed. 43/2 = (43)1/2 = 641/2 = = 8

Simplify an expression that contains a positive rational exponent with a numerator other than 1 In general, we have the following rule. Changing from Rational Exponents to Radicals If m and n are positive integers, x  0, and is in simplified form, then We can interpret xm/n in two ways: 1. xm/n means the mth power of the nth root of x. 2. xm/n means the nth root of the mth power of x.

Example 4 Simplify each expression. a. b. c. Solution: a. or

Example 4 – Solution cont’d b. or c. or

Simplify an expression that contains a negative rational exponent 3.

Simplify an expression that contains a negative rational exponent To be consistent with the definition of negative integer exponents, we define x–m/n as follows. Definition of x–m/n If m and n are natural integers, is in simplified form, and x1/n is a real number (x  0), then and

Example 5 Write each expression without negative exponents. a. b. c. d. (– 16)– 3/4 Solution: a. b.

Example 5 – Solution cont’d c. d. (– 16)– 3/4 is not a real number, because (– 16)– 1/4 is not a real number.

Simplify an expression that contains a negative rational exponent Comment A base of 0 raised to a negative power is also undefined, because 0–2 would equal to which is undefined since we cannot divide by 0.

Simplify an expression that contains rational Simplify an expression that contains rational exponents by applying the properties of exponents 4.

Example 6 Write all answers without negative exponents. Assume that all variables represent positive numbers. Thus, no absolute value symbols are necessary. a. 52/753/7 = 52/7+3/7 = 55/7 b. (52/7)3 = 5(2/7)(3) = 56/7 Apply the rule xmxn = xm + n. Add: Apply the rule (xm)n = xmn. Multiply:

Example 6 cont’d c. (a2/3b1/2)6 = (a2/3)6(b1/2)6 = a12/3b6/2 = a4b3 d. = a8/3+1/3 –2 = a8/3+1/3 – 6/3 = a3/3 = a Apply the rule (xy)n = xnyn. Apply the rule (xm)n = xmn twice. Simplify the exponents. Apply the rules xmxn = xm+n and

Simplify a radical expression by first writing Simplify a radical expression by first writing it as an expression with a rational exponent 5.

We can simplify many radical expressions by using the following steps. Simplify a radical expression by first writing it as an expression with a rational exponent We can simplify many radical expressions by using the following steps. Using Rational Exponents to Simplify Radicals 1. Write the radical expression as an exponential expression with rational exponents. 2. Simplify the rational exponents. 3. Write the exponential expression as a radical.

Example 8 Simplify. Assume variables represent positive values. a. b. c. Solution: a. Apply the rule Write using radical notation.

Example 8 – Solution b. cont’d Apply the rule Write using radical notation.

Example 8 – Solution c. cont’d Write 27 as 33 and write the radical as an exponential expression. Raise each factor to the power by multiplying the fractional exponents. Simplify each fractional exponent. Apply the rule (xy)n = xnyn. Write using radical notation.