Finding Negative Exponents in Algebra
(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 letter a with an exponent of −n. We put the whole power under 1 and remove the minus sign from the exponent. This is equal to 1an.

a to the minus n equals 1 over 2 to the n This is a laws of exponents.

How to Find Negative Exponents in Algebra

Question

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

3

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

1 over x to the 2

Answer:

We have found the negative exponent.

1 over x squared

Understanding Finding a Negative Exponent in Algebra

Let us look at the rule for negative exponents in algebra:

bases and exponents
  • a-−n and an are powers.
  • The bases of the powers are a.
  • The exponent of a−n is −n and the exponent of an is n.
  • The fraction 1an is the reciprocal of an.

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, a2 means that a is multiplied by itself 2 times:
a2 = a × a

What Is a Reciprocal?

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

reciprocals of letters

Reciprocals with Coefficients

What if there is a number or other letter written in front of a power with a negative exponent?

2 x to the minus 2 The number or letter in front is a coefficient that is multiplying what comes after it. It goes on top of the fraction instead of 1.

2 over x squared
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See Also

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