# 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

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

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