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How to Divide Powers (Mathematics Lesson)

Dividing Powers

Powers can be divided.

To divide powers, subtract the exponents from each other.

2 to the m divided by 2 to the n equals 2 to the m minus n

This is a law of exponents.

How to Divide Powers

Dividing powers is easy.

An Example Question

Use the law of exponents to divide the powers below.

2 to the 5 times 2 to the 3
Step 1

Check that the bases of the powers are the same. In our example, the bases are both 2.

2 is the base of both powers
Step 2

Find the exponent of the first power. In our example, the first power has an exponent of 5.

5

Find the exponent of the second power. In our example, the second power has an exponent of 3.

3
Step 3

Subtract the exponents from Step 2 (5 and 3) from each other.

5 - 3 = 2
Step 4

Make the answer from Step 3 (2) the exponent of the base of the powers that have been divided.

2 to the 2

We have divided the powers from each other.

2 to the 5 divided by 2 cubed equals 2 squared

Understanding Dividing Powers

Let us look at the rule for dividing powers:

bases and exponents
  • We are dividing powers. 2m, 2n and 2m - n are powers.

  • The base in each power is 2. This law of exponents only applies when the bases are the same.

  • The exponents in each power are m, n and m - n. This law of exponents applies even when the exponents are different.

Dividing Powers As a Fraction

A division can be written as a fraction.

2 to the m divided by 2 to the n equals 2 to the m over 2 to the n

A Real Example of How to Divide Powers

The slider below shows another real example of how to divide powers.

Interactive Test
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Here's a second test on the dividing powers.
Here's a third test on dividing powers.

Beware

The Bases Must Be The Same

The law of exponents discussed here only works when the bases are the same.

The division below cannot be simplified, and must be left as it is:

Top Tip

0, 1 and Negative Exponents

When subtracting exponents, don't worry if the resulting exponent is 0, 1 or negative.

The relevant laws of exponents are:

  • 20 = 1

  • 21 = 2

  • 2 - n = 1 / 2n