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How to Find Negative Exponents (Mathematics Lesson)

Finding Negative Exponents

A power with a negative exponent is equal to the reciprocal of the power (1 over the power) with the exponent made positive.

Imagine we have the number 2 with an exponent of -n. We put the whole power under 1 and remove the minus sign from the exponent. This is equal to 12n.

This is a laws of exponents.

How to Find Negative Exponents

An Example Question

Use the law of exponents to find the power with the negative exponent below.

3 to the minus 2
Step 1

Write 1 on top of a fraction (called the numerator).

1 over the line
Step 2

Write the power from the question on the bottom of the fraction (called the denominator).

1 over 3 to the minus 2
Step 3

Remove the minus sign from the exponent. In our example, the exponent of -2 becomes 2.

1 over 3 to the 2

We have used the law of exponents to express the power with a negative exponent as a power with a positive exponent.

1 over 3 squared

As the final step, evaluate the power with the positive exponent.

Step 4

Evaluate the power on the bottom of the fraction. In our example, evaluate 32.

32 = 3 × 3 = 9

3-2 is equal to 19.

3 to the minus 2 equals one ninth

Understanding Finding a Negative Exponent

Let us look at the rule for negative exponents:

bases and exponents
  • 2-n and 2n are powers.

  • The bases of the powers are 2.

  • The exponent of 2-n is -n and the exponent of 2n is n.

  • The fraction 12n is the reciprocal of 2n.

Real Example of How to Find Negative Exponents

The slider below shows another real example of how to find negative exponents.

Interactive Test
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Here's a second test on finding negative exponents.
Here's a third test on finding negative exponents.

Note

What Is an Exponent?

An exponent tells you how many times a number or letter is multiplied by itself.

An exponent is denoted by a raised number by the right hand side of the number (called the base) that is multiplied by itself.

For example, 22 means that 2 is multiplied by itself 2 times:

22 = 2 × 2

What Is a Reciprocal?

A reciprocal of a quantity (such as a letter or power) is 1 divided by the quantity.

reciprocals of numbers