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. 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.
Reading Math When a radical sign shows no index, it represents a square root.
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.
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.
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
8 4 x 4 3 27 4 2 x 3 Check It Out! Example 2b Simplify the expression. Assume that all variables are positive. Quotient Property. Rationalize the numerator. Product Property. Simplify.
3 9 x Check It Out! Example 2c Simplify the expression. Assume that all variables are positive. 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
Writing Math The denominator of a rational exponent becomes the index of the radical.
Write the expression (–32) in radical form and simplify. 3 5 ( ) 3 - 5 32 - 32,768 5 Example 3: Writing Expressions in Radical Form Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (–2)3 Evaluate the root. Evaluate the power. –8 Evaluate the power. –8 Evaluate the root.
1 3 64 ( ) 1 ( ) 3 64 1 64 3 3 64 Check It Out! Example 3a Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write will a radical. (4)1 Evaluate the root. Evaluate the power. 4 Evaluate the power. 4 Evaluate the root.
5 2 4 ( ) 5 ( ) 2 4 5 4 2 2 1024 Check It Out! Example 3b Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluatethe power first. Write with a radical. Write with a radical. (2)5 Evaluate the root. Evaluate the power. 32 Evaluate the power. 32 Evaluate the root.
2 1 3 9 3 2 4 4 5 10 81 5 Check It Out! Example 4 Write each expression by using rational exponents. a. b. c. 103 Simplify. 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. Simplify. 72 Evaluate the Power. 49 CheckEnter the expression in a graphing calculator.
1 1 3 3 (–8)– 1 – 1 –8 2 Check It Out! Example 5b Simplify each expression. Negative Exponent Property. 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.
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? Example 6: Chemistry Application
500 (200–) = 500(2– ) 1 1 1 800 t 2 2 1600 1600 2 = 500(2– ) = 500() 1 500 2 2 = Example 6 Continued Substitute 800 for t. Simplify. Negative Exponent Property. Simplify. Use a calculator. ≈ 354 The amount of radium-226 left after 800 years would be about 354 mg.
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.