7.1/7.2 Nth Roots and Rational Exponents How do you change a power to rational form and vice versa? How do you evaluate radicals and powers with rational exponents? How do you solve equations involving radicals and powers with rational exponents?

Objectives/Assignment Evaluate nth roots of real numbers using both radical notation and rational exponent notation. Use nth roots to solve real-life problems such as finding the total mass of a spacecraft that can be sent to Mars.

The Nth root Index Number Radicand Radical The index number becomes the denominator of the exponent. n > 1

Radicals If n is odd – one real root. If n is even and a > 0 Two real roots a = 0 One real root a < 0 No real roots

Example: Radical form to Exponential Form Change to exponential form. or

Example: Exponential to Radical Form Change to radical form. The denominator of the exponent becomes the index number of the radical.

Example: Evaluate Without a Calculator Evaluate without a calculator.

Ex. 2 Evaluating Expressions with Rational Exponents A. B. Using radical notation Using rational exponent notation. OR

Example: Solving an equation Solve the equation: Note: index number is even, therefore, two answers.

Ex. 4 Solving Equations Using nth Roots A. 2x 4 = 162B. (x – 2) 3 = 10

Ex. 1 Finding nth Roots 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: or

Ex. 3 Approximating a Root with a Calculator Use a graphing calculator to approximate: SOLUTION: First rewrite as. Then enter the following: To solve simple equations involving x n, isolate the power and then take the nth root of each side.

Ex. 5: Using nth Roots in Real Life 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 (in kilograms) of the magnetic sail, f is 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 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 = s 3 A volunteer crew of the Olympias was able to generate a maximum power of about 10.5 kilowatts. What was their greatest speed?

SOLUTION The greatest speed attained by the Olympias was approximately 7 knots (about 8 miles per hour).

Rules Rational exponents and radicals follow the properties of exponents. Also, Product property for radicals Quotient property for radicals Quotient property for radicals

Review of Properties of Exponents from section 6.1 a m * a n = a m+n (a m ) n = a mn (ab) m = a m b m a -m = These all work for fraction exponents as well as integer exponents.

Ex: Simplify. (no decimal answers) a.6 1/2 * 6 1/3 = 6 1/2 + 1/3 = 6 3/6 + 2/6 = 6 5/6 b. (27 1/3 * 6 1/4 ) 2 = (27 1/3 ) 2 * (6 1/4 ) 2 = (3) 2 * 6 2/4 = 9 * 6 1/2 c.(4 3 * 2 3 ) -1/3 = (4 3 ) -1/3 * (2 3 ) -1/3 = 4 -1 * 2 -1 = ¼ * ½ = 1 / 8 ** All of these examples were in rational exponent form to begin with, so the answers should be in the same form!

Try These!

Writing Radicals in Simplest Form

Example: Using the Quotient Property Simplify.

Adding and Subtracting Radicals Two radicals are like radicals, if they have the same index number and radicand Example Addition and subtraction is done with like radicals.

Example: Addition with like radicals Simplify. Note: same index number and same radicand. Add the coefficients.

Example: Subtraction Simplify. Note: The radicands are not the same. Check to see if we can change one or both to the same radicand. Note: The radicands are the same. Subtract coefficients.

Writing variable expressions in simplest form