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§ 7.2 Rational Exponents Rational Exponents The Definition of If n a represents a real number and a 1/ n n n 2 a T 1/ n is an integer, then a. If a is negative, n must be odd. If a is nonnegative, n can be any index. P 499 Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.2 Rational Exponents EXAMPLE Use radical notation to rewrite each expression. Simplify, if possible: 1 1 1 4 (a) 3 xy 5 (b) 100 2 (c) 64 3 . SOLUTION (a) 3 xy 4 1 5 5 3 xy 4 1 (b) 100 2 100 10 1 (c) 64 3 3 64 4 Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.2 Rational Exponents Check Point 1 on page 499 1 (a) 25 2 1 (b) 8 3 (c) 5 xy 2 1 4 25 5 8 2 3 4 5 xy 2 Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.2 Rational Exponents EXAMPLE Rewrite with rational exponents: (a) 5 13 x 5 (b) x . SOLUTION Parentheses are needed in part (a) to show that the entire radicand becomes the base. 1 (a) (b) 5 13 x 13 x 5 x x 5 5 1 2 x 5 1 5 2 x2 Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.2 Rational Exponents Check Point 2 on p 500 1 (a) 4 5 xy 5 xy 4 1 3 (b) 5 a b 2 a b 5 2 3 Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.2 Rational Exponents The Definition of m a T m/n If a represents a real number, is a positive rational n number reduced to lowest terms, and n 2 is an integer, then n a m/n a m/n and a m n n m a . Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.2 Rational Exponents EXAMPLE Use radical notation to rewrite each expression and simplify: 3 2 (b) 27 3 . (a) 25 2 SOLUTION 3 (a) 25 2 2 3 25 (b) 27 3 3 5 125 27 3 2 3 9 2 Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.2 Rational Exponents Check Point 3 on p 501 4 (a) 8 3 8 3 (c) - 81 4 16 4 3 4 3 81 27 Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.2 Rational Exponents EXAMPLE Rewrite with rational exponents: (a) 7 x 4 (b) 3 11 xy . SOLUTION 4 (a) (b) 7 x 4 x7 3 11 xy 3 11 xy 2 Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.2 Rational Exponents Check Point 4 on p 501 4 3 (a) (b) 5 6 63 4 7 2 xy 7 2 xy 5 Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.2 Rational Exponents The Definition of If a m/n is a nonzero real number, then a m / n 1 a m/n Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.2 . a m /n T Rational Exponents Check Point 5 on p 502 1 2 (a) 100 1 100 (b) 8 3 5 (d) 3 xy 1 83 2 1 1 8 3 5 9 1 32 10 2 1 3 (c) 32 1 1 1 5 1 5 3 xy 9 Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.2 Rational Exponents in Application EXAMPLE The Galapagos Islands, lying 600 miles west of Ecuador, are famed for their extraordinary wildlife. The function 1 f x 29 x 3 models the number of plant species, f (x), on the various islands of the Galapagos chain in terms of the area, x, in square miles, of a particular island. Use the function to solve the following problem. How many species of plants are on a Galapagos island that has an area of 27 square miles? Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.2 Rational Exponents in Application CONTINUED SOLUTION Because we are interested in how many species of plants there are on a Galapagos island having an area of 27 square miles, substitute 27 for x. Then calculate f (x). 1 f x 29 x 3 This is the given formula. 1 f 27 29 27 3 f 27 29 Replace x with 27. 1 3 27 Rewrite 27 3 as 3 27 . f 27 29 3 Evaluate the cube root. f 27 87 Multiply. A Galapagos island having an area of 27 square miles contains approximately 87 plant species. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.2 Rational Exponents p 502 Properties of Rational Exponents If m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then 1) b b b m n 2) b m b n b mn When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base. mn When dividing exponential expressions with the same base, subtract the exponents. Use this difference as the exponent of the common base. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.2 Rational Exponents p 502 CONTINUED Properties of Rational Exponents If m and n are rational exponents, and a and b are real numbers for which the following expressions are defined, then 3) b b 4) ab n a b 5) a a n b b m n n When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses. mn n n n When a product (not sum) is raised to a power, raise each factor to that power and multiply. When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.2 Rational Exponents EXAMPLE 1 3 Simplify: (a) x7 1 x 7 4 (b) x y 5 1 2 3 3 (c) 1 54 52 1 . 54 SOLUTION 3 (a) x7 1 x 3 x7 7 2 x7 1 7 To divide with the same base, subtract exponents. Subtract. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.2 Rational Exponents CONTINUED 1 1 2 (b) x 4 y 5 1 1 3 1 x4 3 2 y 5 1 x 12 y 3 2 To raise a product to a power, raise each factor to the power. Multiply: 1 1 2 1 2 and . 4 3 12 5 3 15 15 1 1 x 12 Rewrite with positive exponents. 2 y 15 3 (c) 1 3 54 52 54 1 54 1 54 1 3 2 54 1 54 2 5 4 54 1 5 54 1 4 4 54 5 5 54 Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.2 1 Rational Exponents Check Point 6 on page 503 1 1 (a) 7 2 7 3 72 1 1 3 3 76 Simplify: 2 5 6 76 1 (b) 1 50 x 3 5x3 4 10 x 4 3 5x 1 x 3 3 (c) 9 . 1 5 2 4 6 9 . 1 20 5 3 9 . 1 10 1 1 (d) 3 1 x 5y4 3 3 1 x 15 y 12 y 12 1 x5 Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.2 Rational Exponents Simplifying Radical Expressions Using Rational Exponents 1) Rewrite each radical expression as an exponential expression with a rational exponent. 2) Simplify using properties of rational exponents. 3) Rewrite in radical notation if rational exponents still appear. Blitzer, Intermediate Algebra, 5e – Slide #21 Section 7.2 Rational Exponents EXAMPLE Use rational exponents to simplify: (a) 6 ab 2 3 2 a b (b) 5 3 2x . SOLUTION (a) 6 3 2 a b 1 ab a b ab 2 1 2 6 1 1 2 2 2 6 2 1 a 6b 6 a 3b 3 1 4 1 1 a 6 a 6b 3b 3 1 a6 3 a a6 b 4 6 b Rewrite as exponential expressions. 1 2 1 1 3 3 1 1 3 b3 Raise each factor in parentheses to its related power. To raise powers to powers, multiply. Reorder the factors. To multiply with the same base, add exponents. Blitzer, Intermediate Algebra, 5e – Slide #22 Section 7.2 Rational Exponents 5 CONTINUED 2 a 6b 3 5 4 a b 6 a b (b) 5 3 6 2x 5 Add. 5 Rewrite exponents with common denominators. 6 4 5 a b 1 Factor 1/6 out of the exponents. 6 4 1 2 x 3 1 1 5 2 x 3 1 2 x 15 15 2x Rewrite in radical notation. Write the radicand as an exponential expression. Write the entire expression in exponential form. To raise powers to powers, multiply the exponents. Rewrite in radical notation. Blitzer, Intermediate Algebra, 5e – Slide #23 Section 7.2 Rational Exponents Important to Remember: • An expression with rational exponents is simplified when no parentheses appear, no powers are raised to powers, each base occurs once, and no negative or zero exponents appear. • Some radical expressions can be simplified using rational exponents. Rewrite the expression using rational exponents, simplify, and rewrite in radical notation if rational exponents still appear. Blitzer, Intermediate Algebra, 5e – Slide #24 Section 7.2 DONE Rational Exponents Rational exponents have been defined in such a way so as to make their properties the same as the properties for integer exponents. In this section we explore the meaning of a base raised to a rational (fractional) exponent. We will also discover how we can use rational exponents to simplify radical expressions. Blitzer, Intermediate Algebra, 5e – Slide #26 Section 7.2 Rational Exponents Important to Remember: Blitzer, Intermediate Algebra, 5e – Slide #27 Section 7.2