Chapter 7 – Powers, Roots, and Radicals 7.2 – Properties of Rational Exponents

 Today we will learn how to:  Use properties of radicals and rational exponents

7.2 – Properties of Rational Exponents  Product and Quotient Properties of Radicals  Product Property  n √a  b = n √a  n √b  Quotient Property  n √a/b = n √a/ n √b

7.2 – Properties of Rational Exponents  Example 1  3 √125  3 √8  5 √96/ 5 √3

7.2 – Properties of Rational Exponents  Simplest form – When there are no perfect nth powers in the radicand and no radicals in any denominators.  Apply the properties of radicals, remove any perfect nth powers, and rationalize the denominator to write radicals in simplest form.

7.2 – Properties of Rational Exponents  Example 2  3 √104  3 √(1/32)

7.2 – Properties of Rational Exponents Properties of Rational Exponents Let a and b be positive real numbers. Let m and n be rational numbers. Property a m  a n = a m + n (a m ) n = a mn (ab) m = a m b m a -m = 1/a m, a ≠ 0 a m /a n = a m – n (a/b) m = a m /b m, b ≠ 0 Example 3 1/2  3 3/2 = 3 (1/2 + 2/3) = 3 2 = 9 (2 3/2 ) 2 = 2 (3/2  2) = 2 3 = 8 (9  4) 1/2 = 9 1/2  4 1/2 = 3  2 = /2 = 1/16 1/2 = ¼ 5 3/2 /5 1/2 = 5 (3/2 – ½) = 5 1 = 5 (8/27) 1/3 = 8 1/3 /27 1/3 = 2/3

7.2 – Properties of Rational Exponents  Example 3  7 3/4  7 1/4  (6 1/4 ) 4  (49  16) 1/2  64 1/3  8 3/2 /8 1/2

7.2 – Properties of Rational Exponents  Like Radicals – radicals with the same index and same radicand.  3 √2 and 4 3 √2  To add or subtract like radicals, use the distributive property.

7.2 – Properties of Rational Exponents  Example 4  7 5 √12 – 5 √12  4(9 2/3 ) + 8(9 2/3 )

7.2 – Properties of Rational Exponents  Simplifying Variable Expressions  You can apply the properties of radicals to expressions involving variables.  When the value of a variable is negative, you may need to use absolute value to find an nth root.  n √x n = x when n is odd  n √x n = |x|  Absolute is not needed when all variables are assumed to be positive. In this lesson, assume that all variables are positive.

7.2 – Properties of Rational Exponents  Example 5  Simplify the expression. Write your answer using positive exponents only. Assume all variables are positive.  √16x 4  (16y 8 ) 1/4  3 √(x 6 /y 9 )  4y 5/4 z/yz -3

7.2 – Properties of Rational Exponents  Example 6  Simplify the expression. Assume all variables are positive.  5√y – 2√y  6x 2 y 3/4 + 3x 2 y 3/4

7.2 – Properties of Rational Exponents HOMEWORK 7.2 Worksheet