# 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 = 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 × am = an + 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 = a4 + 7 = a11

Let's do
58 / 55

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 = 53

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

Let's do also
715 / 79

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 = 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.

Law #2:
am / an
=  am - n

What about
79 / 715

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
=  am - n

79 / 715
=  79 - 15   =   7-6

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 = 83 + 3 + 3 + 3 = 812

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

Law #3: (an)m = an × 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

## Recent Articles 1. ### Basic Math Review Game - Math adventure!

May 19, 19 09:20 AM

Basic math review game - The one and only math adventure game online. Embark on a quest to solve math problems!

Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons. Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

## Recent Articles 1. ### Basic Math Review Game - Math adventure!

May 19, 19 09:20 AM

Basic math review game - The one and only math adventure game online. Embark on a quest to solve math problems!

Read More

Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.