5 6 Radical Expressions and Rational Exponents Warm

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5 -6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt.

5 -6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal ALgebra 2 Algebra 2 Holt

5 -6 Radical Expressions and Rational Exponents Warm Up Simplify each expression. 1. 73

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Objectives Rewrite radical expressions by using rational

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Vocabulary index rational exponent Holt Mc. Dougal

5 -6 Radical Expressions and Rational Exponents Vocabulary index rational exponent Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents The nth root of a real number

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Reading Math When a radical sign shows

5 -6 Radical Expressions and Rational Exponents Reading Math When a radical sign shows no index, it represents a square root. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 1: Finding Real Roots Find all

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 1 Find all

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents The properties of square roots in Lesson

5 -6 Radical Expressions and Rational Exponents The properties of square roots in Lesson 1 -3 also apply to nth roots. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Remember! When an expression contains a radical

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 2 A: Simplifying Radical Expressions Simplify

5 -6 Radical Expressions and Rational Exponents Example 2 A: Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Factor into perfect fourths. Product Property. 3 x x x 3 x 3 Holt Mc. Dougal Algebra 2 Simplify.

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 2 a Simplify

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 2 a Simplify the expression. Assume that all variables are positive. 4 4 16 x 4 4 24 • x 4 2 x 2 x Holt Mc. Dougal Algebra 2 Factor into perfect fourths. Product Property. Simplify.

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 2 c Simplify

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 2 c Simplify the expression. Assume that all variables are positive. 3 x 9 x 3 Holt Mc. Dougal Algebra 2 Product Property of Roots. Simplify.

5 -6 Radical Expressions and Rational Exponents A rational exponent is an exponent that

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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Writing Math The denominator of a rational

5 -6 Radical Expressions and Rational Exponents Writing Math The denominator of a rational exponent becomes the index of the radical. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 3: Writing Expressions in Radical Form

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 Mc. Dougal 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 3 a 1

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 3 a 1 3 Write the expression 64 in radical form, and simplify. Method 1 Evaluate the root first. ( 64 ) 1 3 (4)1 4 Write with a radical. Evaluate the root. Evaluate the power. Holt Mc. Dougal Algebra 2 Method 2 Evaluate the power first. 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 3 b Write

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 3 b 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 Mc. Dougal 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 3 c Write

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 3 c Write the expression 625 simplify. Method 1 Evaluate the root first. ( 4 625 ) 3 Evaluate the root. 125 Evaluate the power. Holt Mc. Dougal 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

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 am = a Simplify. m n 3 15 5 33 27 Holt Mc. Dougal Algebra 2 n am = a m n Simplify.

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 4 Write each

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 1000 Holt Mc. Dougal Algebra 2 5 Simplify. 5 2 4 1 2 Simplify.

5 -6 Radical Expressions and Rational Exponents Rational exponents have the same properties as

5 -6 Radical Expressions and Rational Exponents Rational exponents have the same properties as integer exponents (See Lesson 1 -5) Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 5 A: Simplifying Expressions with Rational

5 -6 Radical Expressions and Rational Exponents Example 5 A: 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 Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 5 B: Simplifying Expressions with Rational

5 -6 Radical Expressions and Rational Exponents Example 5 B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. 1 4 Holt Mc. Dougal Algebra 2 Evaluate the power.