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.

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! NO Radicals in the denominator!

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. Or convert to rational exponent 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 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

Example 3: Writing Expressions in Radical Form Write the expression (–32) in radical form and simplify. Two methods: 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.

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   stop

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.

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)?

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.