2. Aim: Rational Exponents Course: Adv. Alg. & Trig. Aim: How do we handle fractional exponents? Do Now: 2 8 = 2 ? = 2 6 = 2 ? = 2 2 = 2 1 = 2 ? = 256 128 64 32 2 5 = 16 4 2 4 = 2 7 = 2 1/4 1/8 2 -2 = 2 -3 = Fill in the appropriate information 2 3 = 8 2 -1 = 1/2

3. Aim: Rational Exponents Course: Adv. Alg. & Trig. Aim: How do we handle fractional exponents? Do Now: Simplify/Rationalize:

4. Aim: Rational Exponents Course: Adv. Alg. & Trig. Properties of Exponents Zero Power Property a 0 = 1 Product of Powers Property a m a n = a m+n Power of Power Property (a m ) n = a mn Negative Power Property a -n = 1/a n, a  0 Power of Product Property (ab) m = a m b m Quotients of Powers PropertyPower of Quotient Property

5. Aim: Rational Exponents Course: Adv. Alg. & Trig. Indices, Exponents, and New Power Rules Product of Powers Property a m a n = a m+n Power of Product Property (ab) m = a m b m Power of Quotient Property example: 8 2 8 3 = 8 2 + 3 = 8 5 example: (2 8) 2 = 2 2 8 2 example: x 3 x 6 = x 3 + 6 = x 9 example: (xy) 5 = x 5 y 5 example:

6. Aim: Rational Exponents Course: Adv. Alg. & Trig. Types of exponents Positive Integer Exponent a n = a a a a n factors Rational Exponent 1/n Rational Expo. m/n beware! -2 3/2 is not the same as (-2) 3/2 Negative Exponent -n Zero Exponent a 0 = 1 0 - m/n Negative Rational Exponent

7. Aim: Rational Exponents Course: Adv. Alg. & Trig. Roots, Radicals & Rational Exponents Square Root radical sign radicand index of a number is one of the two equal factors whose product is that number Every positive real number has two square roots The principal square root of a positive number k is its positive square root,. has an index of 2 can be written exponentially as If k < 0, is an imaginary number

8. Aim: Rational Exponents Course: Adv. Alg. & Trig. Cube Root Roots, Radicals & Rational Exponents radical sign radicand index of a number is one of the three equal factors whose product is that number has an index of 3 principal cube roots can be written exponentially as k 1/3

9. Aim: Rational Exponents Course: Adv. Alg. & Trig. Roots, Radicals & Rational Exponents radical sign radicand index n th Root of a number is one of n equal factors whose product is that number can be written exponentially as k 1/n principal odd roots principal even roots has an index where n is any counting number

10. Aim: Rational Exponents Course: Adv. Alg. & Trig. square root - n th root - = k 1/n cube root - Indices and Rational Exponents k 1/2 k 1/2 = k 1/2 + 1/2 = k 1 = k k 1/3 k 1/3 k 1/3 = k 1/3 + 1/3 + 1/3 = k 1 = k k 1/n k 1/n k 1/n... = k 1/n + 1/n + 1/n... = k 1 = k n times 2 1/2 2 1/2 = 2 1/2 + 1/2 = 2 1 = 2 2 1/3 2 1/3 2 1/3 = 2 1/3 + 1/3 + 1/3 = 2 1 = 2 8 1/3 8 1/3 8 1/3 = 8 1/3 + 1/3 + 1/3 = 8 1 = 8

11. Aim: Rational Exponents Course: Adv. Alg. & Trig. Fractional Exponents Radicals Fractional Exponents (ab) m = a m b m

12. Aim: Rational Exponents Course: Adv. Alg. & Trig. rational exponentpositive integer exponent multiplication law Simplifying = x 6 = x 3 division law simplify & power law = x 5 - 2 = x 10 - 4

13. Aim: Rational Exponents Course: Adv. Alg. & Trig. Simplifying – Fractional Exponents A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions. Conditions for a Simplified Expression 1.It has no negative exponents. 2.It has no fractional exponents in the denominator. 3.It is not a complex fraction. 4.The index of any remaining radical is as small as possible.

14. Aim: Rational Exponents Course: Adv. Alg. & Trig. Model Problems Rewrite using radicals: Rewrite using rational exponents: Evaluate:

15. Aim: Rational Exponents Course: Adv. Alg. & Trig. Evaluating Evaluate a 0 + a 1/3 + a -2 when a = 8 8 0 + 8 1/3 + 8 -2 replace a with 8 1 + 8 1/3 + 8 -2 x 0 = 1 x 1/3 = 1 + 2 + 8 -2 x –n = 1/x n 8 –2 = 1/8 2 = 1/64 1 + 2 + 1/64 3 1/64 combine like terms If m = 8, find the value of (8m 0 ) 2/3 (8 8 0 ) 2/3 replace m with 8 (8) 2/3 (8 1) 2/3 x 0 = 1 = 4

16. Aim: Rational Exponents Course: Adv. Alg. & Trig. Simplifying – Fractional Exponents