1. 7.1 nth Roots and Rational
Exponents

2. What is an “nth Root?”
 Extends the concept of square roots.
 For example:
◦ A cube root of 8 is 2, since 23 = 8
◦ A fourth root of 81 is 3, since 34 = 81
 For integers n greater than 1, if bn = a
then b is an nth root of a.
 Written where n is the index of the
radical.

3. Rational Exponents
nth roots can be written using rational
exponents.
 For example:
 In general, for any integer n
greater than 1.

4. Real nth Roots
 If n is odd:
◦ a has one real nth root

 If n is even:
◦ And a > 0, a has two real nth roots

◦ And a = 0, a has one nth root, 0

◦ And a < 0, a has no real nth roots


5. Finding nth Roots
Find the indicated real nth root(s) of a.
 Example: n = 3, a = -125
 n is odd, so there is one real cube root: (-
5)3 = -125
 We can write

6. Example:
 n = 4, a = 16
 n is even, and a > 0, so 16 has two real 4th
roots: 24 = 16 and (-2)4 = 16
 We can write

7. Your Turn!
 Find the indicated real nth root(s) of a.
 n = 4, a = 625
 n = 3, a = -27

8. Rational Exponents
 Not always of the form 1/n.


 The denominator of the exponent is the
index of the radical.

9. Evaluating with Rational Exponents
 Evaluate:
 93/2
 32-2/5

10. Your Turn!
 Evaluate:
 493/2
 16-3/4

11. Approximating with a Calculator
 Rewrite in rational exponent notation, then
use calculator.
 Example:
 Use ( ) around the exponent!
 Your Turn!
 Approximate

12. Solving Equations with nth Roots
 Get the power alone, then take nth roots
of both sides.
 Examples:
2x4 = 162 (x – 2)3 = 10

13. Your Turn!
 Solve each equation.
 5x4 = 80
 (x – 1)3 = 32