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 = 2³

We call 2 the base and 3 the exponent.

Let us now try to perform the following multiplication:

2³ × 2²

2³ × 2² = (2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 = 2⁵

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

3 + 2 = 5

In the same way,

4³ × 4⁴ = (4 × 4 × 4) × (4 × 4 × 4 × 4)= 4⁷

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

Law #1: aⁿ × a^m = a^{n + m}

If a stands for any number, a × a × a × a = a⁴

By the same token,

If a stands for any number, a × a × a × a × a × a × a = a⁷

a⁴ × a⁷ = a^{4 + 7} = a¹¹





Rewrite the problem:





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 = 5³

Notice that you can get the same answer if you do 8 - 5 = 3





Rewrite the problem:





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 = 7⁶

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.





Try now (8³)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 = 8^{3 + 3 + 3 + 3} = 8¹²

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

Law #3: (aⁿ)^m = a^{n × m}

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

The last makes it easy to simplify (6⁵)²⁰⁰

Just multiply 5 and 200 to get 1000 and the answer is 6¹⁰⁰⁰

This figure summarizes the laws of exponents