Negative Exponents

Negative exponents are a mathematical concept used to represent the reciprocal of a number raised to a positive exponent. They are an important part of algebra and are used to simplify expressions and solve equations.

Definition of Negative Exponents
For any non-zero number a and a positive integer n, a negative exponent is defined as:

\(a^{-n} = \frac{1}{a^n} \)

This means that a negative exponent indicates the reciprocal of the base raised to the positive exponent.

Take note:
A negative exponent does not make the value negative. It indicates a reciprocal.

How to rewrite negative exponents?
When the exponent of a number is negative we can rewrite the number with a positive exponent using the negative exponent rule.
The following diagram shows how to rewrite a number with a negative exponent. Scroll down the page for more examples and solutions on negative exponents.

Exponent Worksheets
Practice your skills with the following exponent worksheets:
Printable & Online Exponents Worksheets

Free Exponent Games Online
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We can also rewrite:

\( \frac{1}{a^{-n}} = a^{n} \)

Also, for a fraction:

\( ( \frac{a}{b})^{-n} = ( \frac{b}{a} )^n \)

Key Points:

• Non-zero base: The base ‘a’ cannot be zero, because division by zero is undefined.
• Reciprocal: A negative exponent tells you to take the reciprocal of the base raised to the positive version of that exponent. It does not make the number negative.
• Location in a fraction: A term with a negative exponent in the numerator moves to the denominator (and its exponent becomes positive). A term with a negative exponent in the denominator moves to the numerator (and its exponent becomes positive).

Example:
Rewrite the following with positive exponents:

Solution:

Examples of negative exponents

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Negative exponents
Learn how to solve negative exponents in these step by step examples

• Show Step-by-step Solutions

Exponent Rules, Negative Exponents

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