Understanding Rational Exponents and Nth Roots
Learn about rational exponents, nth roots, even and odd exponents, evaluating expressions, rules for rational exponents, simplifying radicals, and more in this informative content with visual examples.
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How Do We Use Rational Exponents? Do Now: Perform the indicated operation and simplify 1. 2. 1
nth Roots nth Roots An nth root of number a is a number whose nth power is a. a number whose nth power is a = na If the index n is even, then the radicand a must be nonnegative. is not a real number 5 32 2 = = 4 4 16 2, 16 but 2
Square Root of x2 x = 2 x 3 Page 393
Radicals 4
When n Is Even 1 = = 100 100 10 2 1 = = 4 625 625 5 4 1 = = 6 64 64 2 6 1 ( ) = 4 4 is not yet defined 2 7
Exponent 1/n When n Is Odd 1 = = 3 27 27 3 3 1 ( ) = = 3 27 27 3 3 1 1 1 1 5 = = 5 32 32 2 9
nth Root of Zero 0 = n 0 10
Evaluating in Either Order ( ) 8 2 ( ) 8 or ( ) 2 2 2 = = = 3 4 3 2 ( ) 8 = = = 2 3 3 8 64 4 3 12
Evaluating a-m/n 1 1 1 1 2 ( ) 8 = = = = ( ) 8 3 ( ) 2 2 2 2 4 ( ) 8 3 3 14
Rules for Rational Exponents 15 7-6
Simplifying ( ) a 1 = = 6 6 y y y 6 6 1 1 ( )= ab b 3 2 16
Simplifying ( ) y 1 = = 6 6 y y 6 6 1 1 1 1 ( ) ( ) = 1 1 a b ab a b a b 3 3 2 2 1 1 + + 1 1 = a b 3 2 2 3 = a b 3 2 17
Simplifying ( ) a 1 = = 6 6 y y y 6 6 1 2 1 3 ( ) = b ab a b 3 3 2 2 ( ) = 1 8 10 12 9 x y z 2 18
Multiplying Radicals Different Indices 1 1 1 1 3 + = = = = = 3 4 4 4 2 2 2 2 2 2 2 8 4 2 4 2 4 = 3 2 3 19
Multiplying Radicals Different Indices 1 1 1 1 3 + = = = = = 3 4 4 4 2 2 2 2 2 2 2 8 4 2 4 2 4 1 1 = = 3 2 3 2 3 3 2 20
Different Indices 1 1 1 1 3 + = = = = = 3 4 4 4 2 2 2 2 2 2 2 8 4 2 4 2 4 1 2 3 1 = = = 3 2 3 2 3 2 3 3 6 6 2 21
Different Indices 1 1 1 1 3 + = = = = = 3 4 4 4 2 2 2 2 2 2 2 8 4 2 4 2 4 1 2 3 1 = = = = 6 2 3 6 3 2 3 2 3 2 3 2 3 3 6 6 2 22
Different Indices 1 1 1 1 3 + = = = = = 3 4 4 4 2 2 2 2 2 2 2 8 4 2 4 2 4 1 2 3 1 = = = = 6 2 3 6 3 6 2 3 2 3 2 3 2 3 108 3 6 6 2 23
Rational Exponents Eliminate the root, then the power 2 = 2 a 3 24
Eliminate the Root, Then the Power 2 = 2 a 3 3 2 = 3 2 a 3 = 2 8 a = 2 8 a = 2 2 a CHECK 25
Negative Exponents 2 ( ) = 1 1 r 3 26
Negative Exponents Eliminate the root, then the power ( ) r 1 = 2 1 3 3 2 ( ) = 3 1 1 r 3 ( ) 2 = 1 1 r ( ) 2 = 1 1 r = 1 1 r = = 2 0 r r 27 CHECK
Negative Exponents Eliminate the root, then the power 2 ( ) = 2 3 1 t 3 28
No Solution Eliminate the root, then the power 2 ( ) = 2 3 1 t 3 3 2 ( ) ( ) 3 = 2 3 1 t 3 ( ) 2 = 2 3 1 t ( ) 2 = 2 3 1 t 29
No Solution Eliminate the root, then the power 2 ( ) = 2 3 1 t 3 3 2 ( ) ( ) 3 = 2 3 1 t 3 ( ) 2 = 2 3 1 t ( ) 2 = 2 3 1 t No real solution 30
Strategy for Solving Equations with Exponents and Radicals 31
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