Laws of exponents
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
Let's do also
7^{15}
/
7^{9}
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.
Law #2:
a^{m}
/
a^{n}
= a
^{m - n}
What about
7^{9}
/
7^{15}
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
a^{m}
/
a^{n}
= a
^{m - n}
7^{9}
/
7^{15}
= 7
^{9 - 15} = 7
^{-6}
Try now (8
^{3})
^{4}
An important observation:
In (
8^{3})
^{4}, the blue part is the base now and 4 is the exponent
Therefore, you can multiply
8^{3} by itself 4 times.
8^{3} ×
8^{3} ×
8^{3} ×
8^{3} =
8^{3 + 3 + 3 + 3} =
8^{12}
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}