Exponents Powers and Exponents

9/13/2013 Exponents 2 Integer Powers For any real number a, how can we represent a a or a a a ? By convention, we write a a = a 2 and a a a = a 3 In general, for any positive integer n, the n th power of a is a a … a Powers n factors = a n= a n

9/13/2013 Exponents 3 Integer Powers Examples: 3 3 = 3 2 = 9 (–3) (–3) = (–3) 2 = = 2 5 = 32 Powers

9/13/2013 Exponents 4 The n th Power For any real number a, and positive integer n we write the n th power of a as Powers and Exponents a na n Base a Exponent n Base 3 Exponent  E Example: 4 th power of 3

9/13/2013 Exponents 5 Combining Exponents Consider the following: Exponents = (3 3) (3 3 3) = factors of 3 = factors of 3 3 factors of 3 Notice that 5 = Is this significant ?

9/13/2013 Exponents 6 Combining Exponents Let’s try Exponents = (3) (3 3 3) 3 factors of 3 1 factor of 3 = factors of 3 = 3 4 Question: For positive integers m and n consider 3 m 3 n = 3 r What can be said about r ?

9/13/2013 Exponents 7 Addition of Exponents: General Rule To multiply powers of the same base, add the exponents: a m a n = a m + n  For what bases and exponents does this work?  As demonstrated, this works for positive base and positive integer exponents Exponents

9/13/2013 Exponents 8 Addition of Exponents: General Rule a m a n = a m + n Exponents 16 = = = = = 1/2 = /4 = /8 = 2 -3 OR 81 = = = = = 1/3 = /9 = /27 = 3 -3 OR  C Consider: ? ?

9/13/2013 Exponents 9 Negative Exponents: General Rule We have seen that Exponents = = a –m a ma m 1 = 2 1 = = = = General Rule: = = = = 3 1 = = OR

9/13/2013 Exponents 10 Addition of Negative Exponents What is ? Consider Exponents = = 3 -5 = = = for m, n both positive OR both negative a m a n = a m + n  G General Rule:

9/13/2013 Exponents 11 Positive and Negative Exponents What is ? Exponents = 3 –1 = = = = =  C C Consider: = = = = = =

9/13/2013 Exponents 12 Positive and Negative Exponents What is ? Exponents =  G G General Rule: for any integers m, n a m a –n = a m – n

9/13/2013 Exponents 13 The Zero Exponent We saw that 2 0 = 1 and 3 0 = 1 Is this true in general for any a 0 ? For any non-zero a and any integer m Exponents amam amam = = 1 Undefined !! a 0 = a m – m Then what is 0 0 ?  W If a = 0 ? a 0 = 1

9/13/2013 Exponents 14 Division of Powers For non-zero base a and any integers m and n, what is Power Functions amam anan = amam anan 1 = amam a –n = a m – n amam anan ?  R R Rewriting:

9/13/2013 Exponents 15 Division of Powers Power Functions amam anan = amam anan 1 = amam a –n = a m – n amam anan =  F F For non-zero base a and integers m, n  G General Division Rule:

9/13/2013 Exponents 16 Examples: Rewrite the following in simplified form: 1. Power Functions x7x7 x4x4 x x x x x x x x x = (x x x x) = (x x x) 1 = = x3x3 (x x x x) (x x x) (x x x x) =

9/13/2013 Exponents 17 Examples: Rewrite the following in simplified form: Power Functions x7x7 x4x4 = x 7 – 4 = x3x3 x4x4 x7x7 = x 4 – 7 = x –3

9/13/2013 Exponents 18 Powers of Powers Example: 8 2 = (2 3 ) 2 = = (2 2 2) (2 2 2) = = 2 6 = 2 32 Power Functions

9/13/2013 Exponents 19 Powers of Powers In General: Suppose we have b n and told that b = a m How do we find b n as a power of a ? This is just a power of a power: b n = ( a m ) n = a m n Power Functions

9/13/2013 Exponents 20 Mixed Bases and Powers Examples: 1. ( a b ) 3 = ( a b ) ( a b ) ( a b ) = ( a a a ) ( b b b ) = a 3 b 3 Power Functions

9/13/2013 Exponents 21 Mixed Bases and Powers Examples: 2. (x 2 y 4 ) 3 = (x 2 ) 3 (y 4 ) 3 = x 6 y 12 Does this work if, from 1., we have a = x 2 and b = y 4 ? Power Functions Question:

9/13/2013 Exponents 22 Mixed Bases and Powers Examples: 3. Power Functions x2x2 y2y2 = = x y x y = x x y y 2 ( x y )

9/13/2013 Exponents 23 Mixed Bases and Powers Examples: Power Functions = x3x3 y5y5 x3x3 y5y5 = x6x6 y 10 = a8x6a8x6 b –4 y 10 = a8b4x6a8b4x6 y 10 2 ( x3x3 y5y5 ) = x3x3 y5y5 x3x3 y5y5 2 ( a4x3a4x3 b –2 y 5 )

9/13/2013 Exponents 24 Mixed Bases and Powers General Rules: 1. For any real numbers a and b and any integer n ( a b ) n = a n b n 2. For any real numbers a and b, with b non-zero, and any integer n Power Functions = b n a n ( a b n )

9/13/2013 Exponents 25 Power Functions Definition: A power function f(x) is defined by f(x) = x b for some constant b Examples: f(x) = x 2 f(x) = x –4 f(x) = x 2/3 Power Functions

9/13/2013 Exponents 26 Radical Functions Definition: A radical function f(x) is defined by f(x) = x b for b = 1/n for some integer n ≥ 2 Examples: f(x) = x 1/2 f(x) = x 1/3 Power Functions  x = x 3  =

9/13/2013 Exponents 27 Rational Exponent Equations Solve: 1. x 1/4 = x 1/3 – 5 = 1 3. n –2 + 3n –1 + 2 = 0 Power Functions

9/13/2013 Exponents 28 Think about it !