F RACTIONAL E XPONENTS AND R ADICALS Expressing powers with rational exponents as radicals and vice versa

T ODAY ’ S O BJECTIVES Students will be able to demonstrate an understanding of powers with integral and rational exponents 1. Explain, using patterns, why a 1/n = n √a, n > 0 2. Express powers with rational exponents as radicals and vice versa 3. Identify and correct errors in a simplification of an expression that involves powers

M ULTIPLYING P OWERS We have learned in the past that: a m *a n = a m+n We can extend this law to powers with fractional exponents with numerator 1: 5 1/2 * 5 1/2 = 5 1/2 +1/2 =5 1 =5 And: √5 * √5 = √25 = 5 So, 5 1/2 and √5 are equivalent expressions

M ULTIPLYING P OWERS Similarly, 5 1/3 * 5 1/3 * 5 1/3 = 5 1/3+1/3+1/3 = 5 1 = 5 And: 3 √5 * 3 √5 * 3 √5 = 3 √125 = 5 So, 5 1/3 = 3 √5 These examples indicate that: Raising a number to the exponent ½ is equivalent to taking the square root of the number Raising a number to the exponent 1/3 is equivalent to taking the cube root of the number, and so on

P OWERS WITH R ATIONAL E XPONENTS WITH N UMERATOR 1 When n is a natural number and x is a rational number: x 1/n = n √x

E XAMPLE 1: E VALUATING P OWERS OF THE FORM A 1/ N Evaluate each power without using a calculator: 27 1/3 27 1/3 = 3 √27 = / /2 = √0.49 = 0.7 Your turn: (-64) 1/3 (4/9) 1/2 -4, 2/3

P OWERS WITH DECIMAL EXPONENTS A fraction can be written as a terminating or repeating decimal, so we can interpret powers with decimal exponents; for example: 0.2 = 1/5, so = 32 1/5 What about when the numerator is greater than 1? To give meaning to a power such as 8 2/3, we extend the exponent law (a m ) n = a mn so that it applies when m and n are rational numbers.

N UMERATORS GREATER THAN 1 We write the exponent 2/3 as 1/3 * 2, or as 2 * 1/3 So: 8 2/3 = (8 1/3 ) 2 = ( 3 √8) 2 Take the cube root of 8, then square the result 3 √8 = 2, 2 2 = 4 Or: 8 2/3 = (8 2 ) 1/3 = 3 √8 2 Square 8, then take the cube root of the result 8 2/3 = 3 √64 = 4

P OWERS WITH RATIONAL EXPONENTS These examples illustrate that the numerator of a fractional exponent represents a power and the denominator represents a root. The root and power can be evaluated in any order When m and n are natural numbers, and x is a rational number: x m/n = (x 1/n ) m = ( n √x) m And: x m/n = (x m ) 1/n = n √x m

E XAMPLE 2: R EWRITING P OWERS IN R ADICAL AND E XPONENT F ORM Write 40 2/3 in radical form in 2 ways Use a m/n = ( n √a) m or n √a m 40 2/3 = ( 3 √40) 2 or 3 √40 2 Your turn: Write √3 5 and ( 3 √25) 2 in exponent form 3 5/2 and 25 2/3

R EVIEW Powers with Rational Exponents with Numerator 1 When n is a natural number and x is a rational number: x 1/n = n √x Powers with rational exponents When m and n are natural numbers, and x is a rational number: x m/n = (x 1/n ) m = ( n √x) m And: x m/n = (x m ) 1/n = n √x m

H OMEWORK Pg , 5, 7, 9, 11, 15, 17-21