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Laws of Exponents

​
m ≥ 0
2
n ≥ 0
5
laws, assuming a, b > 0:
m
a
n
a
m
a
n
a
​
m
a
n
)\),
a b
m
)\),
m
a
b
0
a
-1
a
-m
a
1
m
a
2
a
×
5
a
=
2 times
a × a
×
5 times
a × a × a × a × a
​ =
7 times
a × a × a × a × a × a × a
​ =
7
a
The laws of exponents show the power of elementary algebra and lay the groundwork for logarithms. The laws are:
m
a
n
a
=
m+n
a
m
a
n
a
=
m-n
a
n
(
m
a
)
=
mn
a
m
(ab)
=
m
a
n
b
m
a
b
=
m
a
m
b
0
a
=1
-1
a
=
1
a
-m
a
=
1
m
a
1
m
a
=
m
a
This Demonstration does not prove the first five of these laws; for a proof, you need to use mathematical induction.
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