Laws of exponents help us to simplify terms containing exponents. We derive these laws here using some good examples

Recall that 2 × 2 × 2 = 2

We call 2 the base and 3 the exponent.

Let us now try to perform the following multiplication:

2

2

Notice that we can get the same answer by adding the exponents

3 + 2 = 5

In the same way,

4

In general, add exponents to multiply numbers with the same base

If a stands for any number, a × a × a × a = a

By the same token,

If a stands for any number, a × a × a × a × a × a × a = a

a

Let's do

5^{8}
5^{5}

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

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

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.

a^{m}
a^{n}

= a
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
^{m - n}

a^{m}
a^{n}

= a
7^{9}
7^{15}

= 7Try now (8

In (8

Therefore, you can multiply 8

8

Notice that you can get 12 by multiplying 3 and 4 since 3 × 4 = 12

All the laws of exponents are very useful, especially the last one.

The last makes it easy to simplify (6

Just multiply 5 and 200 to get 1000 and the answer is 6