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 × 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

This figure summarizes the laws of exponents

Laws of exponents

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

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