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Defining, Rewriting, and Evaluating Rational Exponents Adapted from Walch Education Concepts: • Rational exponents are another way to write radical expressions. 1 x = x and 2 1 3 x = x3 . • the rules and properties that apply to integer exponents also apply to rational exponents. 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 2 Concepts, cont. Key Concepts • An exponential expression contains a base and a power. • A base is the quantity that is being raised to a power. • A power, also known as an exponent, is the quantity that shows the number of times the base is being multiplied by itself in an exponential expression. In the exponential expression an, a is the base and n is the power. 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 3 More… • A radical expression contains a root, which can be shown using the radical symbol, . • The root of a number x is a number that, when multiplied by itself a given number of times, equals x. And more… • The root of a function is also referred to as the inverse of a power, and “undoes” the power. 3 8 = 2 and 23 = 8. • In the radical expression n an , the nth root of the nth power of a is a. • Roots can be expressed using a rational exponent instead of the radical symbol. 1 m n a = a and n 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents n x = xn. m 5 Continued… • A rational exponent is an exponent that is a rational number. • A rational number is any number that can be m written as n , where both m and n are integers and n ≠ 0. • The denominator of the rational exponent is the root, and the numerator is the power. mn n m a = a . 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 6 Properties of Exponents Words Symbols Zero Exponent Property A base raised to the power of 0 is equal to 1. a0 = 1 Numbers 120 = 1 Negative Exponent m Property 2 1 1 1 n 3 a = , A negative exponent of a 64 = 2 = m 16 number is equal to the n 3 a 64 reciprocal of the positive exponent of the number. a ¹ 0, n ¹ 0 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 7 Properties of Exponents Words Symbols Product of Powers m n m+n a · a = a Property To multiply powers with the same base, add the exponents. Quotient of Powers Property To divide powers with the same base, subtract the exponents. Numbers 1 7 1 34 · 34 = 34 + 7 4 = 32 = 9 4 a m a n =a m-n 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 89 1 =8 4 1 9 9 1 = 83 = 2 89 8 Properties of Exponents Words Symbols Power of a Power Property To raise one power to another power, multiply the exponents. Power of a Product Property To find the power of a product, distribute the exponent. Numbers (a ) m n = am · n 3 2 æ ö ·3 3 3 = ç5 ÷ = 5 è ø 2 52 = 25 ( ab) m = amb m 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents ( 25 · 36) 1 1 2 = 1 25 2 · 36 2 = 5 · 6 = 30 9 Properties of Exponents Words Symbols Power of a Quotient Property To find the power of a quotient, distribute the exponent. m Numbers æ aö a ç b ÷ = bm è ø m 1 1 æ 25 ö 2 25 2 5 ç 49 ÷ = 1 = 7 è ø 49 2 • Either the power or root can be determined first when evaluating an exponential expression with a rational exponent. 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 10 One more thing… • Sometimes rational exponents appear as decimals. 1 x0.25 is equal to x 4 or 4 x. 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 11 For Example… • How can the expression rational exponent? 8 ac be rewritten using a 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 12 Steps: 1. Identify the numerator of the rational exponent. • The numerator is the power: c. 2. Identify the denominator of the rational exponent. • The denominator is the root: 8. 3. Rewrite the expression in the form , where the base is the quantity raised to a power and of which the root is being taken. c 8 ac = a 8 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 13 Your Turn • Evaluate the exponential expression 4 1 3 2 æ ö 3 ç ÷ è ø Round your answer to the nearest thousandth. 4.1.1: Defining, Rewriting, and Evaluating Rational Exponents 14 Dr. Dambreville THANKS FOR WATCHING!