Laws of exponents
The laws of exponents help us to simplify terms containing exponents. We derive these laws here using some good examples.
A little reminder before we derive these laws of exponents:
Recall that 2 × 2 × 2 = 2
3
We call 2 the base and 3 the exponent.
Let us now try to perform the following multiplication:
2
3 × 2
2
2
3 × 2
2 = (2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 = 2
5
Notice that we can get the same answer by adding the exponents.
3 + 2 = 5
In the same way,
4
3 × 4
4 = (4 × 4 × 4) × (4 × 4 × 4 × 4)= 4
7
In general, add exponents to multiply numbers with the same base.
Law #1: a
n × a
m = a
n + m
If a stands for any number, a × a × a × a = a
4
By the same token,
If a stands for any number, a × a × a × a × a × a × a = a
7
a
4 × a
7 = a
4 + 7 = a
11
We get
5 × 5 × 5 × 5 × 5 × 5 × 5 × 5
/
5 × 5 × 5 × 5 × 5
Rewrite the problem:
We get
5 × 5 × 5 × 5 × 5
/
5 × 5 × 5 × 5 × 5
× 5 × 5 × 5
Notice that
5 × 5 × 5 × 5 × 5
/
5 × 5 × 5 × 5 × 5
= 1
The reason for this is that whenever you divide something by the same thing, the answer is always 1.
The problem becomes 1 × 5 × 5 × 5 = 5 × 5 × 5 = 5
3
Notice that you can get the same answer if you do 8 - 5 = 3
We get
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
/
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
Rewrite the problem:
We get
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
/
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
× 7 × 7 × 7 × 7 × 7 × 7
Notice once again that
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
/
7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
= 1
The reason for this is that whenever you divide something by the same thing, the answer is always 1.
The problem becomes 1 × 7 × 7 × 7 × 7 × 7 × 7 = 7
6
Notice that you can get the same answer if you do 15 - 9 = 6
In general, when dividing with exponents, you can just subtract the exponent of the denominator from the exponent of the numerator.
It is the same problem as before. However, this this time 9 is on top and 15 is at the bottom.
We can just use the formula
am
/
an
= a
m - n
Try now (8
3)
4
An important observation:
In (
83)
4, the blue part is the base now and 4 is the exponent
Therefore, you can multiply
83 by itself 4 times.
83 ×
83 ×
83 ×
83 =
83 + 3 + 3 + 3 =
812
Notice that you can get 12 by multiplying 3 and 4 since 3 × 4 = 12
Law #3: (a
n)
m = a
n × m
All the laws of exponents are very useful, especially the last one.
The last makes it easy to simplify (6
5)
200
Just multiply 5 and 200 to get 1000 and the answer is 6
1000
This figure summarizes the laws of exponents
Test your knowledge of the laws of exponents with the quiz below
-
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