§ 7.2 Rational Exponents

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 #2 Section 7.2

Rational Exponents The Definition of T If represents a real number and is an integer, then If a is negative, n must be odd. If a is nonnegative, n can be any index. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.2

Rational Exponents EXAMPLE Use radical notation to rewrite each expression. Simplify, if possible: SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.2

Rational Exponents Rewrite with rational exponents: EXAMPLE Rewrite with rational exponents: SOLUTION Parentheses are needed in part (a) to show that the entire radicand becomes the base. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.2

Rational Exponents The Definition of T If represents a real number, is a positive rational number reduced to lowest terms, and is an integer, then and Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.2

Rational Exponents EXAMPLE Use radical notation to rewrite each expression and simplify: SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.2

Rational Exponents Rewrite with rational exponents: EXAMPLE SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.2

Rational Exponents The Definition of T If is a nonzero real number, then Blitzer, Intermediate Algebra, 5e – Slide #9 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 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 #10 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). This is the given formula. Replace x with 27. Rewrite as . Evaluate the cube root. Multiply. A Galapagos island having an area of 27 square miles contains approximately 87 plant species. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.2

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) When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base. 2) 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 #12 Section 7.2

Properties of Rational Exponents 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) 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. 4) When a product (not sum) is raised to a power, raise each factor to that power and multiply. 5) 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 #13 Section 7.2

Rational Exponents Simplify: EXAMPLE Simplify: SOLUTION To divide with the same base, subtract exponents. Subtract. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.2

Rational Exponents CONTINUED To raise a product to a power, raise each factor to the power. Multiply: Rewrite with positive exponents. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.2

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 #16 Section 7.2

Rational Exponents Use rational exponents to simplify: EXAMPLE Use rational exponents to simplify: SOLUTION Rewrite as exponential expressions. 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 #17 Section 7.2

Rational Exponents Add. Rewrite exponents with common denominators. CONTINUED Add. Rewrite exponents with common denominators. Factor 1/6 out of the exponents. 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 #18 Section 7.2

Rational Exponents Important to Remember: Properties of integer exponents are true for rational exponents. Remember those exponent rules we learned before? Well, they still hold, even when the exponents are fractions. 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. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.2

Rational Exponents Important to Remember: 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. Didn’t learn any radical rules? Then rewrite your expression without the radical sign using rational exponents instead. Now you can use your exponent rules to simplify the radical expression. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.2