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Name_____________________________________ Class____________________________ Date________________ Lesson 7-4 (pp. 379–384) Lesson Objective 1 Simplifying expressions with rational exponents Rational Exponents NAEP 2005 Strand: Algebra Topic: Variables, Expressions, and Operations Local Standards: ____________________________________ Key Concepts. Rational Exponents If the nth root of a is a real number and m is an integer, then © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. 1 an n m n a and a n m m n a (a ) If m is negative, a 0. Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator equals 0. Property Example 1 2 1 12 3 am ? an am n 83 ? 83 5 83 (am)n a mn (512)4 512 ? 4 52 25 (ab)m a m b m (4 ? 5)2 42 ? 52 2 ? 52 (am) a 1 m am am n an m a m a b m b () 1 1 5 81 5 8 1 1 1 1 922 912 13 3 p2 32 2 12 π 1 π p 1 p2 ( ) 5 27 1 3 1 1 3 3 5 1 53 273 Another way to write a radical expression is to use a rational exponent. Like the radical form, the exponent form always indicates the principal root. A rational exponent may have a numerator other than 1. All of the properties of integer exponents also apply to rational exponents. You can simplify a number with a rational exponent by using the properties of exponents or by converting the expression to a radical expression. To write an expression with rational exponents in simplest form, write every exponent as a positive number. Algebra 2 Daily Notetaking Guide Lesson 7-4 141 Name_____________________________________ Class____________________________ Date ________________ Examples. 1 Simplifying Expressions With Rational Exponents Simplify each expression. 1 a. 643 1 64 3 3 64 4 3 Rewrite as a radical. 3 Rewrite 64 as a cube. 3 1 cube Definition of root. 1 b. 7 2 ? 7 2 1 7 ? 7 Rewrite as radicals. 7 c. 5 5 1 3 1 3 All rights reserved. 1 72 ? 72 By definition,7 is the number whose square is 7 . 1 25 3 ? 1 3 5 ? 3 5 ? ? 25 3 3 Rewrite as radicals. 25 25 Property for multiplying radical expressions. 3 5 By definition, 5 is the number whose cube is 5 . 2 Converting To and From Radical Form x 2 7 7 x 2 0.4 in radical form. 7 or (x) 2 y 0.4 y 2 5 1 1 or 5 y b. Write the radical expressions c 4 3 3 c 4 3 3 (5 y) 2 2 5 and (b) in exponential form. 5 c 4 (3 b)5 b 3 3 Simplifying Numbers With Rational Exponents Simplify 252.5. Method 1 25 2.5 5 ( 25 2 5 2 5 142 Method 2 Lesson 7-4 5 1 5 5 ( ( ) 5 2 5 2 5 2 25 2.5 5 25 2 1 3125 1 5 5 1 5 25 2 1 (25 ) 5 1 3125 Algebra 2 Daily Notetaking Guide © Pearson Education, Inc., publishing as Pearson Prentice Hall. 2 a. Write x 7 and y Name_____________________________________ Class____________________________ Date ________________ 2 4 Writing Expressions in Simplest Form Write (243a ( ) ( ) 2 2 10 (243a ) 5 (3 5 2 10 a )5 3 5 5 10 5 ) in simplest form. 2 ? a (10) 5 3 2 4 a 3 a 2 9 4 a 4 Check Understanding. 1. Simplify each expression. 1 1 All rights reserved. a. 164 1 3 2. a. Write the expressions y28 and z0.4 in radical form. 1 c. 22 ? 82 2 2 1 5 2 8 3 ; "z " y © Pearson Education, Inc., publishing as Pearson Prentice Hall. 1 b. 22 ? 22 4 3 b. Write the expressions "x2 and (!y) 3 in exponential form. 2 3 x3 ; y2 c. Critical Thinking Refer to the definition of rational exponents. Explain the need for the following restriction: If m is negative, a 0. If m is negative, then a is in the radicand in the denominator of a fraction. If a 0, then the denominator would be zero. Since this cannot happen, a 0. 3. Simplify each number. 232 3 b. 325 a. 25 1 125 4. Write (8x 15) 8 213 4 c. (232) 5 16 in simplest form. 1 2x5 Algebra 2 Daily Notetaking Guide Lesson 7-4 143