5 6 Radical Expressions and Rational Exponents Holt

5 -6 Radical Expressions and Rational Exponents Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents 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. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents 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. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Reading Math When a radical sign shows no index, it represents a square root. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents 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. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents The properties of square roots in Lesson 1 -3 also apply to nth roots. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents 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. NO Radicals in the denominator! Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 2 A: Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Factor into perfect fourths. Product Property. 3 x x x 3 x 3 Holt Mc. Dougal Algebra 2 Simplify. Or convert to rational exponent

5 -6 Radical Expressions and Rational Exponents Example 2 B: Simplifying Radical Expressions Quotient Property. Simplify the numerator. Rationalize the numerator. Product Property. Simplify. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 2 c Simplify the expression. Assume that all variables are positive. 3 x 9 x 3 Holt Mc. Dougal Algebra 2 Product Property of Roots. Simplify.

5 -6 Radical Expressions and Rational Exponents A rational exponent is an exponent that can be expressed as m , where m and n are integers and n n ≠ 0. Radical expressions can be written by using rational exponents. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 3: Writing Expressions in Radical Form 3 5 Write the expression (– 32) in radical form and simplify. Two methods: Method 1 Evaluate the root first. ( -32 ) Write with a radical. (– 2)3 Evaluate the root. – 8 Evaluate the power. 5 Method 2 Evaluate the power first. 3 Holt Mc. Dougal Algebra 2 Write with a radical. 5 -32, 768 Evaluate the power. – 8 Evaluate the root.

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 4 Write each expression by using rational exponents. a. 81 b. 3 4 c. 10 9 3 1000 stop Holt Mc. Dougal Algebra 2 5 Simplify. 5 2 4 1 2 Simplify.

5 -6 Radical Expressions and Rational Exponents Rational exponents have the same properties as integer exponents (See Lesson 1 -5) Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 5 A: 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. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Example 5 B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. 1 4 Holt Mc. Dougal Algebra 2 Evaluate the power.

5 -6 Radical Expressions and Rational Exponents 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)? Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Check It Out! Music Application = 64(2– 1) Use 64 cm for the length of the string, and substitute 12 for n. Simplify. Negative Exponent Property. = 32 Simplify. The fret should be placed 32 cm from the bridge. Holt Mc. Dougal Algebra 2