1. nth Roots and Rational Exponents
Section 5.1 beginning on page 238

2. The Basics 𝑛 𝑥 𝑚 =𝑥 ( 𝑛 𝑥 ) 𝑚 =𝑥 𝑚 𝑛 𝑚 𝑛
𝑛 𝑥 𝑚 =𝑥
Pg. 239 in your textbook
𝑚 𝑛
( 𝑛 𝑥 ) 𝑚 =𝑥
**We will only have two solutions when the radicand is positive and the root is even.**
Pg. 238 in your textbook

3. Finding nth Roots Example 1: Find the indicated real nth root(s) of a.
a) n = 3, a = b) n = 4, a = 81
3 −216
4 81
=−6
=±3
Monitoring Progress: Find the indicated nth root(s) of a.
1) n=4, a=16 2) n=2, a=-49 3) n=3, a=-125 4) n=5, a =243
4 16 =±2
3 −125 =−5
−49 →𝑁𝑜 𝑟𝑒𝑎𝑙 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠
5 243 =3

4. Rational Exponents Example 2: Evaluate each expression
a) b) −3 5
(Write without negative or rational exponents before evaluating the roots and then the powers) =
= 1 ( 5 32 ) 3 =
= 1 8
b) 32 −3 5 a)
= ( 16 ) 3
= 4 3
=64
Monitoring Progress: Evaluate the expression without using a calculator
5) ) 9 − ) )
= 1 243
=32
=27 =1

5. Approximating
Example 3: Evaluate each expressing using a calculator. Round your answer to two decimal places (to the nearest hundredth).
a) b) c) ( 4 7 ) 3
≈1.55
≈2.54
≈4.30
Monitoring Progress: Evaluate the expression using a calculator. Round your answer to two decimal places when appropriate.
9) ) − ) ( 4 16 ) 5 12) ( 3 −30 ) 2
≈2.05
≈0.06
=32
≈9.65

6. Solving Equations Using nth Roots
To solve an equation of the form 𝑢 𝑛 =𝑑, where u is an algebraic expression, take the nth root of each side.
Example 4: Find the real solution(s) of
(a) 4 𝑥 5 =128 (b) (𝑥−3) 4 =21
4 (𝑥−3) 4 = 4 21
𝑥 5 =32
5 𝑥 5 = 5 32
𝑥−3=± 4 21
𝑥=3± 4 21
(exact solution)
𝑥=2 𝑥=
𝑥=3− 4 21
(approximate solutions)
𝑥≈5.14
𝑥≈0.86

7. Solving Equations Using nth Roots
Monitoring Progress: Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate.
13) 8 𝑥 3 = ) 𝑥 5 =512
15) (𝑥+5) 4 = ) (𝑥−2) 3 =−14
𝑥=2
𝑥=4
𝑥=−3 𝑎𝑛𝑑 𝑥=7
𝑥≈−0.41