Exponential Unit Math 11 The expression 2 3 is called a power. The base is 2 , and the exponent is 3. Part A: How can you multiply powers with the same base? Product of Powers Expanded Form Result as a Power (23 )( 24) 2x2x2x2x2x2x2 27 (103 )(105) (-2)2(-2) (x3 )(x3) (-y)2 (-y)3 Part B: How can you divide powers with the same base? Quotient of Powers 75 ÷ 72 103 ÷ 103 (-2)4 ÷ (-2)2 x3 ÷ x 2 (-y)7 ÷ (-7)3 Expanded Form 7x7x7x7x7 7x7 Result as a Power 73 Part C: How can you find the power of a power? Power of a Power (23)2 Expanded Form I Expanded Form II (23 )(23) 2x2x2x2x2x2 Result as a Power 26 (103)4 [(-4)2]2 (x2)3 [(-y)4]2 Part D: How can you find the power of a product? Power of a Product (3 x 2)2 Expanded Form I (3 x 2)(3 x 2) Expanded Form II 3x3x2x2 Result as a Power 32 x 22 (2 x 5)3 (-4 x 3)2 (x y)5 (-p q)3 Part E: How can you find the power of a quotient? Power of a Quotient 3 2 3 3 4 5 x y 4 Expanded Form I 2 2 2 x x 3 3 3 Expanded Form II 2x2x2 3x3x3 Result of a Power 23 33 The Laws of Exponents Law 1 x m x n x m n eg. x5 x 2 x5 2 x7 Law 2 x m x n x mn eg. x5 x 2 x52 x3 x x eg. x x Law 3 m n mn 5 3 xy Law 4 eg. Law 5 n 35 x15 xn y n xy 3 x3 y 3 n x xn yn y y0 5 x x5 eg. 5 y y 1. Simplify a. x5 x 23 b. x12 x 3 x 4 c. m5 m8 m3 d. 9 x5 7 x5 e. 75m18 15m6 f. g. 12 y 9 8 y12 h. 3c 4 j. m11n8 m5 n 4 k. 7c12 d 5 6c8 d 6 15 12 5 4 m. 36a b 9a b n. x 2 y 3 24c8 d 5 15c3d 9 p. 2 5 8c d 18cd i. a 7b 4 a 9b3 5 x y a b q. ab 2 2 3 4 2 3 24a 20 6a10 2 4 l. xy 2 3 12m3n2 18m5 n3 2 2 9mn 4m n o. 2. Evaluate for x 2 and y 1 2 4 x y 3x y 6x y 3 3x y 5x y a. 4 b. 6 2 2 5 7 2 2 3. Simplify 2a 3a 6a 3x a. 2 x 1 4 b. x 2 x x 2m y n 1 3 m 1 yn 3 2 4. If x 3m2 and y m3 , write each expression in terms of m. b. 5x 3 y 4 3 xy a. 2 x 2 y Integral exponents Zero Exponent: Expanding x3 we get x3 x x x x x x which equals 1. Using the law of exponents, we get x3-3 = x0 Therefore, x0=1 (for all x not equal to zero) . Negative exponents: Now, a n Therefore, a 0 n a0 1 n n a a a-n = 1 an x 0 1 where x 0 and x n 1 xn n R, x 0 2 9 3 5. Evaluate without a calculator. 3 a. 2 1 3 b. 4 2 1 c. 2 3 3 d. 4 0 6. Simplify 31 32 a. b. 31 32 4 70 23 2 c. 24c d 4c d 5 1 3 8 3 12m5 n 2 5m11n 6 d. 15m3n 4 Rational Exponents 1 n x n x where n N x 0, if n is even x 1 n 1 where n N x 0, if n is even x n m n x n xm = 7. a. 12 3 5 m x 0, if n is even b. 5 1 17 94 c. 3 25 4 d. 1 52 7 Simplify 2 3 2 a. 25 8 3 9 4 n Express as a power 8. d. x 2 2 2 b. 32 3 2435 1 e. 9 8 1 2 2 3 2 1 c. 64 3 1253 1 4 1 3 f. 125 3 32 5 36 2 2 9. Express as a power with base 2 a. 8 b. 1 c. d. 16 n 2 f. 83n 42 n c. 2 4 n 3 g. n 1 16 e. 2 4 8 h. 4 2 16 n 1 n 3 n 1 n2 n 5 8n 1 10. Express as a power with base 3 a. 4 b. 3 d. 9 4 1 d. 27 32 x 729 c. 812 x 9 e. 3 x 1 2 x n 1