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5-6 5-6 Radical Radical Expressions Expressions and and Rational Rational Exponents Exponents Warm Up Lesson Presentation Lesson Quiz HoltMcDougal ALgebra2Algebra 2 Holt 5-6 Radical Expressions and Rational Exponents Warm Up Simplify each expression. 1. 73 • 72 16,807 118 116 121 3. (32)3 729 4. 75 5 3 20 2 35 7 2. 5. 7 Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Objectives Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Vocabulary index rational exponent Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents The nth root of a real number a can be written as the radical expression , where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Reading Math When a radical sign shows no index, it represents a square root. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Example 1: Finding Real Roots Find all real roots. A. sixth roots of 64 A positive number has two real sixth roots. Because 26 = 64 and (–2)6 = 64, the roots are 2 and –2. B. cube roots of –216 A negative number has one real cube root. Because (–6)3 = –216, the root is –6. C. fourth roots of –1024 A negative number has no real fourth roots. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Check It Out! Example 1 Find all real roots. a. fourth roots of –256 A negative number has no real fourth roots. b. sixth roots of 1 A positive number has two real sixth roots. Because 16 = 1 and (–1)6 = 1, the roots are 1 and –1. c. cube roots of 125 A positive number has one real cube root. Because (5)3 = 125, the root is 5. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents The properties of square roots in Lesson 1-3 also apply to nth roots. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Remember! When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Example 2A: Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Factor into perfect fourths. Product Property. 3xxx 3x3 Holt McDougal Algebra 2 Simplify. 5-6 Radical Expressions and Rational Exponents Check It Out! Example 2a Simplify the expression. Assume that all variables are positive. 4 4 16x 4 4 24 • x4 4 24 •x4 2x 2x Holt McDougal Algebra 2 Factor into perfect fourths. Product Property. Simplify. 5-6 Radical Expressions and Rational Exponents Check It Out! Example 2c Simplify the expression. Assume that all variables are positive. 3 x7 3 3 x2 x9 x3 Holt McDougal Algebra 2 Product Property of Roots. Simplify. 5-6 Radical Expressions and Rational Exponents A rational exponent is an exponent that can be expressed as m , where m and n are integers and n n ≠ 0. Radical expressions can be written by using rational exponents. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Writing Math The denominator of a rational exponent becomes the index of the radical. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Example 3: Writing Expressions in Radical Form 3 5 Write the expression (–32) in radical form and simplify. Method 1 Evaluate the root first. ( -32 ) Write with a radical. (–2)3 Evaluate the root. –8 Evaluate the power. 5 Method 2 Evaluate the power first. 3 Holt McDougal Algebra 2 Write with a radical. 5 -32,768 Evaluate the power. –8 Evaluate the root. 5-6 Radical Expressions and Rational Exponents Check It Out! Example 3a 1 3 Write the expression 64 in radical form, and simplify. Method 1 Evaluate the root first. ( 64 ) Method 2 Evaluate the power first. 1 3 (4)1 4 Write with a radical. Evaluate the root. Evaluate the power. Holt McDougal Algebra 2 3 (64 )1 Write will a radical. 3 64 Evaluate the power. 4 Evaluate the root. 5-6 Radical Expressions and Rational Exponents Check It Out! Example 3b Write the expression 4 simplify. 5 2 Method 1 Evaluate the root first. ( 4) Write with a radical. (2)5 Evaluate the root. 2 32 5 Evaluate the power. Holt McDougal Algebra 2 in radical form, and Method 2 Evaluate the power first. 2 2 (4 )5 Write with a radical. 1024 Evaluate the power. 32 Evaluate the root. 5-6 Radical Expressions and Rational Exponents Check It Out! Example 3c Write the expression 625 simplify. Method 1 Evaluate the root first. ( 4 ) 3 625 Evaluate the root. 125 Evaluate the power. Holt McDougal Algebra 2 in radical form, and Method 2 Evaluate the power first. Write with a radical. (5)3 3 4 4 4 (625 )3 Write with a radical. 244,140,625 Evaluate the power. 125 Evaluate the root. 5-6 Radical Expressions and Rational Exponents Example 4: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. A. B. 13 13 4 8 1 2 n a m =a Simplify. m n 3 15 5 33 27 Holt McDougal Algebra 2 n a m =a m n Simplify. 5-6 Radical Expressions and Rational Exponents Check It Out! Example 4 Write each expression by using rational exponents. a. 81 b. 3 4 c. 10 9 3 103 1000 Holt McDougal Algebra 2 5 Simplify. 5 2 4 1 2 Simplify. 5-6 Radical Expressions and Rational Exponents Rational exponents have the same properties as integer exponents (See Lesson 1-5) Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Example 5A: Simplifying Expressions with Rational Exponents Simplify each expression. Product of Powers. 72 Simplify. 49 Evaluate the Power. Check Enter the expression in a graphing calculator. Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents Example 5B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. 1 4 Holt McDougal Algebra 2 Evaluate the power.