# 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

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}
## This figure summarizes the laws of exponents

## Test your knowledge of the laws of exponents with the quiz below