Finding Negative Exponents
(KS3, Year 7)

The Lesson

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

Question

Use the law of exponents to find the power with the negative exponent below.
3 to the minus 2

Step-by-Step:

1

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

1 over the line

2

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

1 over 3 to the minus 2

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.

4

Evaluate the power on the bottom of the fraction. In our example, evaluate 32.
32 = 3 × 3 = 9

Answer:

We have found the negative exponent.

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.

Lesson Slides

The slider below shows another real example of how to find negative exponents. Open the slider in a new tab

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
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See Also

The laws of exponents What is an exponent? What is a reciprocal? What is a negative number?