8-6 Warm Up Lesson Presentation Lesson Quiz Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt ALgebra2

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.

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 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.

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

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.

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!

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.

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

Simplify the expression. Assume that all variables are positive. Check It Out! Example 2b Simplify the expression. Assume that all variables are positive. 8 4 3 x Quotient Property. Rationalize the numerator. Product Property. 4 2 27 3 x Simplify.

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.

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

The denominator of a rational exponent becomes the index of the radical. Writing Math

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.

( ) ( ) 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.

( ) ( ) 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.

( ) ( ) 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.

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

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

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. 72 Simplify. 49 Evaluate the Power. Check Enter the expression in a graphing calculator.

Example 5B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. 1 4 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.

Simplify each expression. Check It Out! Example 5b Simplify each expression. (–8)– 1 3 1 –8 3 Negative Exponent Property. 1 2 – 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.

Example 6: Chemistry Application 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?

Negative Exponent Property. Example 6 Continued 800 1600 500 (200– ) = 500(2– ) t Substitute 800 for t. 1 2 = 500(2– ) Simplify. = 500( ) 1 2 Negative Exponent Property. = 500 2 1 Simplify. Use a calculator. ≈ 354 The amount of radium-226 left after 800 years would be about 354 mg.

Check It Out! Music Application 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)? 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.

( ) Lesson Quiz: Part I Find all real roots. 1. fourth roots of 625 –5, 5 2. fifth roots of –243 –3 Simplify each expression. 3. 4. 256 y 8 4y2 4 2 5. Write (–216) in radical form and simplify. 2 3 = 36 ( ) - 2 3 216 5 3 21 6. Write using rational exponents. 3 5 21

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