6.1 Evaluate nth Roots and Use Rational Exponents

VOCABULARY

nth root of a

For an integer n greater than 1, if bⁿ = a, then b is an nth root of a.

Index of a radical

An nth root of a is written as , where n is the index of the radical.

REAL nth ROOTS OF a

Let n be an integer (n > 1) and let a be a real number.

If n is an even integer:	If n is an odd integer:
· a < 0 No real nth roots.	· a < 0 One real nth root: =
· a = 0 One real nth root: = _0_	· a = 0 One real nth root: = _0_
· a > 0 Two real nth roots: =	· a > 0 One real nth root: =

Example 1

Find nth roots

Find the indicated real nth root(s) of a.

a. n = 3, a = -64

b. n = 6, a = 729

Solution

a. Because n = 3 is odd and a = -64 _<_ 0, -64 has _one real cube root_. Because

( -4 )³ = -64, you can write = _-4_ or (-64)^1/3 = _-4_.

b. Because n = 6 is even and a = 729 _>_ 0, 729 has _two real sixth roots_. Because

_3⁶_ = 729 and ( -3 )⁶ = 729, you can write = _±3_ or ±729^1/6 = _±3_.

Find the indicated real nth roots of a.

1. n = 4, a = 256

±4

2. n = 3, a = 512

8

RATIONAL EXPONENTS

Let a be a real number, and let m and n be positive integers with n > 1.

a^m/n= (a^1/n)^m = ( ____ )^m

a^-m/n = = = , a ¹ 0

Example 2

Evaluate an expression with rational exponents

Evaluate 8^-4/3.

Solution

Rational Exponent Form	Radical Form
8^-^4/3= = = = _______	8^-^4/3= = _______ = =

Example 3

Solve equations using nth roots

a. 2x⁶ = 1458

x⁶ = _729_

x =

x = _±3_

b. (x + 4)³ = 12

x + 4 =

x = - 4

x » _-1.71_

Example 4

Animal Population The population P of a certain animal species after t months can be modeled by P = C(1.21)^t/3 where C is the initial population. Find the population after 19 months if the initial population was 75.

Solution

P = C(1.21)^t^/3	Write model for population.
= _75(1.21)^19/3_	Substitute for C and t.
» _250.8_	Use a calculator.

The population of the species is about _251_ after 19 months.

Complete the following exercises.

3. Evaluate (-125)^-2/3.

4. Solve (y - 3)⁴ = 200.

+ 3 » 6.76 or

+ 3 » - 0.76

5. The volume of a cone is given by V = , where h is the height of the cone and r is the radius. Find the radius of a cone whose volume is 25 cubic inches and whose height is 6 inches.

1.99 in.