Laws of Exponents

exp onent exp onent (base)  answer or (base)  argument

1) Adding and Subtracting Like Terms - Only terms with like exponents and bases can be added or subtracted.

a) 2a 3  3a 3  a 3  4a 3 b) 2a 3  3a 4 can not collect, NOT like terms

2) Multiplication – When multiplying powers that have the same base, add the exponents.

a) a 2  a 3  a 5 b) (x 2 y 3 z 4 )  (xy 4 )  x3 y 7 z 4

3) Division – When dividing powers that have the same base, subtract the exponents.

a) a 6  a 2  a 4 b) (x 2 y 8 z 4 )  (xy 5 )  xy 3 z 4

4) Power of a Power – When you have a base with more than one exponent, multiple the exponents.

a) (a 6 )2  a12 b) (a m ) n3  a mn3m

5) Power of a Product – When you have multiple bases for a single exponent, the exponent applies to ALL bases.

(2x 2 y 3 ) 4  24 x8 y12 a) (ab)3  a 3b3 b)  16x8 y12

6) Power of a Quotient – When you have a fraction with an exponent, the exponent applies to Numerator and Denominator of the fraction.

3 3  2  23 8  2x 4  23 x12 8x12 a)   b)       3  2 3  6 9 6 9  5  5 125  y z  y z y z

7) Zero Exponent – Any expression with a zero exponent is equal to 1

a) (a)0  1 b) (a)0  1 c)  a 0  1 8) Negative Exponent – You can not leave an answer with a negative exponent. Take the reciprocal of the base and write the exponent as a positive.

2 2 2 2 2 2 4 2 1 1  3   4  16 x y x p a) 3   b)       c) 2 4  2 2 32 9  4   3  9 z p y z

m 9) Fractional Exponents – A base with a fraction for an exponent. We can write b n m  1   n  as b  . This will help us in simplifying a problem.   m The following is also true: b n  n b m

Simplify:

3  3 2  1  4 a) 16 2 b) 125 3 c)    81 3 4 3 2  81  1   1   2   3  3  16   125   1       4   81   43  52   3  64  25  3  27

The following table should be known:

Exponent **** 1 2 3 4 5 6 7 8 9 10 1 1 1 1 1 1 1 1 1 1 1 2 2 4 8 16 32 64 128 256 512 1024 3 3 9 27 81 243 729 2187 X X X 4 4 16 64 256 1024 4096 X X X X Base 5 5 25 125 625 3125 X X X X X 6 6 36 216 1296 X X X X X X 7 7 49 343 2401 X X X X X X 8 8 64 512 4096 X X X X X X 9 9 81 729 6561 X X X X X X 10 10 100 1000 10,000 X X X X X X