7.1 nth Roots and Rational Exponents Algebra 2 Mrs. Spitz Spring 2009 Objectives/Assignment вЂў Evaluate nth roots of real numbers using both radical notation and rational exponent notation. вЂў Use nth roots to solve real-life problems such as finding the total mass of a spacecraft that can be sent to Mars. Assignment: pp. 404-405 #4-61 every 3rd Evaluating Nth Roots вЂў You can extend the concept of a square root to other types of roots. For instance 2 is a cube root of 8 because 23 = 8, and 3 is a fourth root of 81 because 34 = 81. In general, for an integer n greater than 1, if bn = a, then b is an nth root of a. An nth root of a is written n a , where n is the index of the radical. Evaluating Nth Roots вЂў You can also write an nth root of a as a power of a. For the particular case of a square root, suppose that a пЂЅ a k. Then you can determine a value for k as follows: aп‚· a пЂЅa a п‚·a пЂЅ a k a 2k k пЂЅa 1 Definition of a square root Substitute ak for a Product of powers property 2 k пЂЅ 1 Set exponents equal when bases are equal. k пЂЅ 1 2 Solve for k Evaluating Nth Roots вЂў Therefore, you can see that a пЂЅ a a similar way, you can show that 3 n a пЂЅa a пЂЅa 1 3 1 n and 4 a пЂЅa 1 4 1 2 . In . In general, for any integer n greater than 1 Ex. 1 Finding nth Roots вЂў Find the indicated real nth root(s) of a. A. n = 3, a = -125 Solution: Because n = 3 is odd, a = -125 has one real cube root. Because (-5)3 = -125, you can write: 3 пЂ 125 пЂЅ пЂ 5 or ( пЂ 125 ) 1 3 пЂЅ пЂ5 Ex. 1 Finding nth Roots вЂў Find the indicated real nth root(s) of a. B. n = 4, a = 16 Solution: Because n = 4 is even, a = 16 > 0, has two real fourth roots. Because 24 = 16, and (-2)4 = 16, you can write: п‚± 16 пЂЅ п‚± 2 4 or п‚± 16 1 4 пЂЅ п‚±2 A rational exponent does not have to be of the form 1 where n is an integer greater than 1. n Other rational numbers such as can also be used as exponents. 3 2 and пЂ 1 2 Ex. 2 Evaluating Expressions with Rational Exponents A. 9 3 2 пЂЅ ( 9 ) пЂЅ 3 пЂЅ 27 3 3 1 3 Using radical notation Using rational exponent notation. пЂЅ ( 9 ) пЂЅ 3 пЂЅ 27 1 1 1 B. 32 пЂ 2 5 пЂЅ 1 пЂЅ пЂЅ 2 пЂЅ 2 2 5 2 4 32 5 ( 32 ) OR OR 9 2 32 2 пЂ2 5 пЂЅ 3 3 1 1 ( 32 ) 5 2 пЂЅ 1 2 2 пЂЅ 1 4 Study Tip вЂў When using a graphing calculator to approximate an nth root, you may have to rewrite the nth root using a rational exponent. Then use the calculatorвЂ™s power key. Ex. 3 Approximating a Root with a Calculator вЂў Use a graphing calculator to approximate: 4 ( 5) 3 SOLUTION: First rewrite ( enter the following: 4 5) 3 as 5 3 4 . Then To solve simple equations involving xn, isolate the power and then take the nth root of each side. Ex. 4 Solving Equations Using nth Roots A. 2x4 = 162 B. (x вЂ“ 2)3 = 10 2 x пЂЅ 162 (x вЂ“ 2) пЂЅ 10 x пЂЅ 81 x -2 пЂЅ x пЂЅ п‚± 81 xпЂЅ x пЂЅ п‚±3 x п‚» 4 . 15 4 4 4 3 3 3 10 10 пЂ« 2 Ex. 5: Using nth Roots in Real Life вЂў The total mass M (in kilograms) of a spacecraft that can be propelled by a magnetic sail is, in theory, given by: where m is the mass 2 0 . 015 m (in kilograms) of the M пЂЅ magnetic sail, f is fd 4 3 the drag force (in newtons) of the spacecraft, and d is the distance (in astronomical units) to the sun. Find the total mass of a spacecraft that can be sent to Mars using m = 5,000 kg, f = 4.52 N, and d = 1.52 AU. Solution The spacecraft can have a total mass of about 47,500 kilograms. (For comparison, the liftoff weight for a space shuttle is usually about 2,040,000 kilograms. Ex. 6: Solving an Equation Using an nth Root вЂў NAUTICAL SCIENCE. The Olympias is a reconstruction of a trireme, a type of Greek galley ship used over 2,000 years ago. The power P (in kilowatts) needed to propel the Olympias at a desired speed, s (in knots) can be modeled by this equation: P = 0.0289s3 A volunteer crew of the Olympias was able to generate a maximum power of about 10.5 kilowatts. What was their greatest speed? SOLUTION вЂў The greatest speed attained by the Olympias was approximately 7 knots (about 8 miles per hour).