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**5-6 Warm Up Lesson Presentation Lesson Quiz**

Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt ALgebra2

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**Simplify each expression.**

Warm Up Simplify each expression. 1. 73 • 72 16,807 2. 118 116 121 3. (32)3 729 4. 75 5 3 20 7 2 35 7 5.

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**Objectives Rewrite radical expressions by using rational exponents.**

Simplify and evaluate radical expressions and expressions containing rational exponents.

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Vocabulary index rational exponent

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**You are probably familiar with finding the square root of a number**

You are probably familiar with finding the square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers. 5 and –5 are square roots of 25 because 52 = 25 and (–5)2 = 25 2 is the cube root of 8 because 23 = 8. 2 and –2 are fourth roots of 16 because 24 = 16 and (–2)4 = 16. a is the nth root of b if an = b.

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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.

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**When a radical sign shows no index, it represents a square root.**

Reading Math

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**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.

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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.

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**The properties of square roots in Lesson 1-3 also apply to nth roots.**

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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. Remember!

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**Example 2A: Simplifying Radical Expressions**

Simplify each expression. Assume that all variables are positive. Factor into perfect fourths. Product Property. 3 x x x Simplify. 3x3

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**Simplify the expression. Assume that all variables are positive.**

Check It Out! Example 2a Simplify the expression. Assume that all variables are positive. 4 16 x 4 4 24 •4 x4 Factor into perfect fourths. 4 24 •x4 Product Property. 2 x Simplify. 2x

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**Simplify the expression. Assume that all variables are positive.**

Check It Out! Example 2c Simplify the expression. Assume that all variables are positive. 3 9 x Product Property of Roots. x3 Simplify.

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A rational exponent is an exponent that can be expressed as , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents. m n

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**The denominator of a rational exponent becomes the index of the radical.**

Writing Math

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**Example 3: Writing Expressions in Radical Form**

Write the expression (–32) in radical form and simplify. 3 5 Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) - 3 5 32 Write with a radical. Write with a radical. (–2)3 Evaluate the root. Evaluate the power. - 5 32,768 –8 Evaluate the power. –8 Evaluate the root.

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**( ) ( ) Check It Out! Example 3a 64**

1 3 Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) 1 3 64 Write with a radical. ( ) 1 3 64 Write will a radical. (4)1 Evaluate the root. Evaluate the power. 3 64 4 Evaluate the power. 4 Evaluate the root.

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**( ) ( ) Check It Out! Example 3b 4**

5 2 Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) 5 2 4 Write with a radical. ( ) 5 2 4 Write with a radical. (2)5 Evaluate the root. 2 1024 Evaluate the power. 32 Evaluate the power. 32 Evaluate the root.

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**( ) ( ) Check It Out! Example 3c 625**

4 Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) 3 4 625 Write with a radical. ( ) 3 4 625 Write with a radical. (5)3 Evaluate the root. 4 244,140,625 Evaluate the power. 125 Evaluate the power. 125 Evaluate the root.

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**Example 4: Writing Expressions by Using Rational Exponents**

Write each expression by using rational exponents. A. B. 4 8 13 = m n a a = m n a a 15 5 3 13 1 2 Simplify. 33 Simplify. 27

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**Write each expression by using rational exponents.**

Check It Out! Example 4 Write each expression by using rational exponents. a. b. c. 3 4 81 9 3 10 2 4 5 103 Simplify. 5 1 2 Simplify. 1000

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**Rational exponents have the same properties as integer exponents (See Lesson 1-5)**

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**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.

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**Example 5B: Simplifying Expressions with Rational Exponents**

Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. 1 4 Evaluate the power.

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