nth Roots and Rational Exponents Section 5.1 beginning on page 238

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

Finding nth Roots Example 1: Find the indicated real nth root(s) of a. a) n = 3, a = -216 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

Rational Exponents Example 2: Evaluate each expression a) 16 3 2 b) 32 −3 5 (Write without negative or rational exponents before evaluating the roots and then the powers) = 1 32 3 5 = 1 ( 5 32 ) 3 = 1 2 3 = 1 8 b) 32 −3 5 a) 16 3 2 = ( 16 ) 3 = 4 3 =64 Monitoring Progress: Evaluate the expression without using a calculator 5) 4 5 2 6) 9 −5 2 7) 81 3 4 8) 1 7 8 = 1 243 =32 =27 =1

Approximating Example 3: Evaluate each expressing using a calculator. Round your answer to two decimal places (to the nearest hundredth). a) 9 1 5 b) 12 3 8 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) 6 2 5 10) 64 −2 3 11) ( 4 16 ) 5 12) ( 3 −30 ) 2 ≈2.05 ≈0.06 =32 ≈9.65

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=3+ 4 21 𝑥=3− 4 21 (approximate solutions) 𝑥≈5.14 𝑥≈0.86

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 =64 14) 1 2 𝑥 5 =512 15) (𝑥+5) 4 =16 16) (𝑥−2) 3 =−14 𝑥=2 𝑥=4 𝑥=−3 𝑎𝑛𝑑 𝑥=7 𝑥≈−0.41