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What Are the Power Rules for Variables?

It is very important to know the six rules of powers, or exponentiation. Here, we’ll go through each of them and you’ll see why they work. Rules 1 and 2 are presented in this entry, followed by Rules 3 and 4, about dividing powers, and Rules 5 and 6, about powers of parentheses. At the end, you will find some examples where you’ll have to use several of the rules at the same time.

There’s no reason to be afraid of these. As long as you make sure to follow the rules, you’ll always end up in the right place. That’s why one of the first steps you should take is to memorize these six rules! But first, let’s examine what powers look like.

Rule

Powers

The expression ab is called a power.

a is called the base.

b is called the exponent.

Examples of various powers

Rule 1

Rule

To the Power 0

Anything with an exponent of 0 equals 1, as long as a0 (00 is undefined).

a0 = 1

Example 1

Here are a couple of different examples of 0 being the exponent, showing that the answer is always 1.

2750 = 10 = x0 = 5.60 = y0 = 1

Rule 2

Rule

Multiplying Two Powers with the Same Base

am an = am+n

Let’s have a look at how this must be true with the help of an example:

a3 = a a a

and

a4 = a a a a

This means that a3 a4 = a a a a3 a a a aa4 = a a a a a a a = a3+4 = a7.

This rule is very important to memorize. Here are some more examples:

Example 2

Simplify x3 x4 as much as possible

x3 x4 = x3+4 = x7

Simplify a2 a5 as much as possible

a2 a5 = a2+5 = a3

Simplify b7 b13 as much as possible

b7 b13 = b7+(13) = b713 = b20

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How to Divide Powers