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How to Find Negative Exponents in Algebra (Mathematics Lesson)

Finding Negative Exponents in Algebra

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

An Example Question

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

x 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 x to the minus 2
Step 3

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

1 over x to the 2

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.

Real Example of How to Find Negative Exponents in Algebra

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

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Here's a second test on finding negative exponents in algebra.
Here's a third test on finding negative exponents in algebra.

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, 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