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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?

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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.

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

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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

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

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

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Example: Evaluate Without a Calculator Evaluate without a calculator.

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

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

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

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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

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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.

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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.

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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.

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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?

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

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

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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.

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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!

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Try These!

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Writing Radicals in Simplest Form

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Example: Using the Quotient Property Simplify.

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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.

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

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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.

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Writing variable expressions in simplest form

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