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## 8-6

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**8-6**Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt ALgebra2**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**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.**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.**The properties of square roots in Lesson 1-3 also apply to**nth roots.**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 Check It Out! 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 Check It Out! Example 2b Simplify the expression. Assume that all variables are positive. Quotient Property. Rationalize the numerator. Product Property. Simplify.**3**9 x Check It Out! 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 Check It Out! 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 Check It Out! 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 Check It Out! 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 Check It Out! 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 (See Lesson 1-5)**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.**Check It Out! 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 Check It Out! Example 5b Simplify each expression. Negative Exponent Property. Evaluate the Power. Check Enter the expression in a graphing calculator.**Check It Out! Example 5c**Simplify each expression. Quotient of Powers. 52 Simplify. Evaluate the power. 25 Check Enter the expression in a graphing calculator.**Radium-226 is a form of radioactive element that decays over**time. An initial sample of radium-226 has a mass of 500 mg. The mass of radium-226 remaining from the initial sample after t years is given by . To the nearest milligram, how much radium-226 would be left after 800 years? Example 6: Chemistry Application**500 (200–) = 500(2– )**1 1 1 800 t 2 2 1600 1600 2 = 500(2– ) = 500() 1 500 2 2 = Example 6 Continued Substitute 800 for t. Simplify. Negative Exponent Property. Simplify. Use a calculator. ≈ 354 The amount of radium-226 left after 800 years would be about 354 mg.**To find the distance a fret should be place from the bridge**on a guitar, multiply the length of the string by , where n is the number of notes higher that the string’s root note. Where should the fret be placed to produce the E note that is one octave higher on the E string (12 notes higher)? Check It Out! Music Application Use 64 cm for the length of the string, and substitute 12 for n. = 64(2–1) Simplify. Negative Exponent Property. Simplify. = 32 The fret should be placed 32 cm from the bridge.**2**3 5. Write (–216) in radical form and simplify. ( ) 2 = 36 - 3 216 5 3 5 21 21 3 Lesson Quiz: Part I Find all real roots. 1. fourth roots of 625 –5, 5 2. fifth roots of –243 –3 Simplify each expression. 4. 4y2 8 3. 256 y 4 2 6. Write using rational exponents.**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