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5-6. Radical Expressions and Rational Exponents. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 2. Holt ALgebra2. 1. r =. x = 7 and x = - 1 4. d =. 2. no solution. x < -1 or x > 0 -5 < x -3 -3 < x -2 x < 3 OR x > 4 m < 0 or m 4

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**5-6**Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt ALgebra2**1. r=**• x= 7 and x=-1 • 4. d= 2. no solution. • x < -1 or x > 0 • -5 < x -3 • -3 < x -2 • x < 3 OR x > 4 • m < 0 or m 4 • 5 < s < 9 • z -24 or z > 4 • x < -12 or x > 15**118**2. 116 75 5 3 2 35 20 7 7 Warm Up Simplify each expression. 16,807 1. 73•72 121 729 3. (32)3 4. 5.**Objectives**Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents.**Vocabulary**index rational exponent**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.**Reading Math**When a radical sign shows no index, it represents a square root.**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.**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.**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.**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**Example 2B: Simplifying Radical Expressions**Quotient Property. Simplify the numerator. Rationalize the numerator. Product Property. Simplify.**424 •4 x4**424 •x4 You Try! Example 2a Simplify the expression. Assume that all variables are positive. 4 4 16 x Factor into perfect fourths. Product Property. 2 x Simplify. 2x**8**4 x 4 3 27 4 2 x 3 You Try! Example 2b Simplify the expression. Assume that all variables are positive. Quotient Property. Rationalize the numerator. Product Property. Simplify.**3**9 x You Try! Example 2c Simplify the expression. Assume that all variables are positive. Product Property of Roots. x3 Simplify.**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**Writing Math**The denominator of a rational exponent becomes the index of the radical.**Write the expression (–32) in radical form and simplify.**3 5 ( ) 3 - 5 32 - 32,768 5 Example 3: Writing Expressions in Radical Form Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (–2)3 Evaluate the root. Evaluate the power. –8 Evaluate the power. –8 Evaluate the root.**1**3 64 ( ) 1 ( ) 3 64 1 64 3 3 64 You Try! Example 3a Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write will a radical. (4)1 Evaluate the root. Evaluate the power. 4 Evaluate the power. 4 Evaluate the root.**5**2 4 ( ) 5 ( ) 2 4 5 4 2 2 1024 You Try! Example 3b Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluatethe power first. Write with a radical. Write with a radical. (2)5 Evaluate the root. Evaluate the power. 32 Evaluate the power. 32 Evaluate the root.**3**4 625 ( ) 3 ( ) 3 4 625 625 4 244,140,625 4 You Try! Example 3c Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (5)3 Evaluate the root. Evaluate the power. 125 Evaluate the power. 125 Evaluate the root.**4**15 1 2 8 5 A. B. 13 m m = = m m n n a a a a n n 3 13 Example 4: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. Simplify. 33 Simplify. 27**2**1 3 9 3 2 4 4 5 10 81 5 You Try! Example 4 Write each expression by using rational exponents. a. b. c. 103 Simplify. Simplify. 1000**Rational exponents have the same properties as integer**exponents**Example 5A: Simplifying Expressions with Rational Exponents**Simplify each expression. Product of Powers. Simplify. 72 Evaluate the Power. 49 CheckEnter the expression in a graphing calculator.**1**4 Example 5B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. Evaluate the power.**Example 5B Continued**Check Enter the expression in a graphing calculator.**You Try! Example 5a**Simplify each expression. Product of Powers. Simplify. 6 Evaluate the Power. Check Enter the expression in a graphing calculator.**1**1 3 3 (–8)– 1 – 1 –8 2 You Try! Example 5b Simplify each expression. Negative Exponent Property. Evaluate the Power. Check Enter the expression in a graphing calculator.**You Try! Example 5c**Simplify each expression. Quotient of Powers. 52 Simplify. Evaluate the power. 25 Check Enter the expression in a graphing calculator.**Lesson Quiz: Part II**7. If $2000 is invested at 4% interest compounded monthly, the value of the investment after t years is given by . What is the value of the investment after 3.5 years? $2300.01

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