5 6 Radical Expressions and Rational Exponents Warm

5 -6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal ALgebra 2 Algebra 2 Holt

5 -6 Radical Expressions and Rational Exponents Warm Up Simplify each expression. 1. 73 • 72 16, 807 118 116 121 3. (32)3 729 4. 75 5 3 5. 20 2 5 2. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Objectives Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Essential Question • How do you multiply/divide powers with the same base when the exponents are rational? Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Vocabulary index rational exponent Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents You are probably familiar with finding the square root of a number and squaring 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 unless otherwise stated. Reading Math When a radical sign shows no index, it represents a square root (index of 2). 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 (even) 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 (odd) root. Because (– 6)3 = – 216, the root is – 6. C. fourth roots of – 1024 A negative number has no real fourth (even) roots. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents The properties of square roots in Lesson 2 -5 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. 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.

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. The denominator of a rational exponent becomes the index of the radical. 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. 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 3 b Write the expression 4 simplify. 5 2 Method 1 Evaluate the root first. ( 4) Write with a radical. (2)5 Evaluate the root. 2 32 5 Evaluate the power. Holt Mc. Dougal Algebra 2 in radical form, and Method 2 Evaluate the power first. 2 2 (4 )5 Write with a radical. 1024 Evaluate the power. 32 Evaluate the root.

5 -6 Radical Expressions and Rational Exponents Example 4: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. A. B. 13 13 4 8 1 2 n am = a Simplify. m n 3 15 5 33 27 Holt Mc. Dougal Algebra 2 n am = a m n Simplify.

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 3 103 Simplify Holt Mc. Dougal Algebra 2 1000 5 Simplify. 5 2 4 1 2 Simplify.

5 -6 Radical Expressions and Rational Exponents Rational exponents have the same properties as integer exponents. 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 Check It Out! Example 5 a Simplify each expression. Product of Powers. Simplify. 6 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 Example 5 B Continued Check Enter the expression in a graphing calculator. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 5 c Simplify each expression. Quotient of Powers. 52 Simplify. 25 Evaluate the power. Check Enter the expression in a graphing calculator. Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Check It Out! Example 5 b Simplify each expression. (8) 1 8 – 1 3 1 2 Holt Mc. Dougal Algebra 2 Negative Exponent Property. Evaluate the Power.

5 -6 Radical Expressions and Rational Exponents 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. 2 5. Write (– 216) 4 4 y 2 256 y 8 2 3 in radical form and simplify. 2 3 -216 = 36 5 3 5 6. Write 21 using rational exponents. 21 3 Holt Mc. Dougal Algebra 2 ( )

5 -6 Radical Expressions and Rational Exponents Essential Question • How do you multiply/divide powers with the same base when the exponents are rational? Holt Mc. Dougal Algebra 2

5 -6 Radical Expressions and Rational Exponents Q: How is an artificial tree like the fourth root of − 68? A: Neither has real roots. Holt Mc. Dougal Algebra 2