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What Are the Power Rules?

The rules of powers are shown below. It’s important that you learn these. We’ll take a closer look at the rules and see how they’re used.

Rule

Rules for Powers

an am = an+m an am = anm (ab) c = abc (a b)n = an bn (a b)n = an bn a = a1 2 an = 1 an a0 = 1

Remember that the letters a and b can be any number or letter. These rules are valid when calculating with both numbers and letters.

Let’s have a look at some examples where we use the rules for powers.

Rule 1

First, I will show you how to use Rule 1.

Rule

an am = an+m

Example 1

Let’s see why this rule is valid: 34 33 = (3 3 3 3) (3 3 3) = 34+3 = 37

This gives us that

34 33 = 34+3 = 37

Rule 2

Next, we will look at Rule 2.

Rule

(ab) c = abc

Example 2

Let’s see why this rule is valid: (42) 4 = (42) (42) (42) (42) = (4 4) (4 4) (4 4) (4 4) = 4 4 4 4 4 4 4 4 = 48

This means that

(42) 4 = 424 = 48

Rule 3

Now you will see how to use Rule 3.

Rule

(a b)n = an bn

Example 3

Let’s see why this rule is valid: (2 3) 3 = (2 3) (2 3) (2 3) = 2 2 2 3 3 3 = 23 33 = 8 27

That gives us

(2 3) 3 = 23 33 = 8 27

Rule 4

Here you will see how to use Rule 4.

Rule

an am = anm

Example 4

Let’s see why this rule is valid: 34 32 = 3 3 3 3 3 3 = 32

This means that

34 32 = 342 = 32

Rule 5

Now, we will take a look at Rule 5.

Rule

an = 1 an

Example 5

Let’s see why this rule is valid: 72 = 702 = 70 72 = 1 72,

showing us that

72 = 1 72

Rule 6

Now you will see how to use Rule 6.

Rule

(a b)n = an bn

Example 6

Let’s see why this rule is valid: (5 2)4 = (5 2) (5 2) (5 2) (5 2) = (5 5 5 5) (2 2 2 2) = 54 24,

showing that

(5 2)4 = 54 24

Rule 7

Now, let’s look at Rule 7.

Rule

a = a1 2

Example 7

Let’s see why this rule is valid. Here we need to be a bit more thorough. We know that 32 = 3. With this as a starting point, we get 32 = 3 (32) 12 = 312 (3) 212 = 312 (3) 2 2 = 31 2 (3) 1 = 312 3 = 31 2

Rule 8

Finally, I will show you how to use Rule 8.

Rule

a0 = 1

Example 8

Let’s see why this rule is valid. Here we will use a trick, namely that 0 = 1 1. It might seem a bit funny, but as long as it works, which it does, we can use it to help us out. 650 = 6511 = 651 651 = 65 65 = 1

This shows us that

650 = 1

Combined examples

Here are some examples that require you to use several rules at once.

Example 9

(2a)3 (a b)2 = 23 a3 a2 b2 = 8 a3 a2 b2 = 8 a3+2 b2 = 8a5b2

Example 10

a4 (a2 b)5 (a b)3 = a4 (a2)5 b5 a3 b3 = a4 a10 b5 a3 b3 = a14 b5 a3 b3 = a143 b53 = a11 b2

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