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 ) n3 a mn3m
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