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. When a radical sign shows no index, it represents a square root. Even roots MUST be a positive Answer. Reading Math

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

( ) ( ) 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.

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

( ) ( ) 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.

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

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.

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!

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

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.

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

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.

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.