The Rules of Dividing Exponents

By Shannon Hill
The division rules for exponents can help you complete algebraic equations.

In algebra, exponents are the expressions used to show you how many times you must multiply the product by itself. They are also known as "powers." For example, 5 to the third power is 5 X 5 X 5, or 125. There are rules for working with computations that contain exponents, such as multiplication and division.

Same Base When the Numerator is Larger

When dividing like bases with exponents, the exponents must be subtracted. If the numerator is larger, the exponent in the denominator is subtracted from the exponent in the numerator. For example, "x" to the 4th power divided by "x" to the 2nd power would be solved by subtracting the exponent 2 from 4. Thus, the answer would be "x" to the 2nd power.

Same Base When the Denominator is Larger

When the bases are the same, but the denominator is larger, subtract the exponent in the denominator from the exponent in the numerator. This may result in a negative exponent. For example, "x" to the 4th power divided by "x" in the 5th power would be solved by subtracting 5 from 4. The answer is "x" to the -1 power.

When this occurs, you can write the negative exponent as a fraction, making the answer 1 divided by "x" to the 1st power.

More Than One Base and Exponent

When more than one base and exponent is included in the same equation, you must follow the rules for each. For example, "x" to the 3rd power "y" to the 5th power divided by "x" to the 2nd power "y" to the 8th power would be solved by subtracting the exponents with the like base of "x", resulting in "x" and subtracting the exponents with the like base of "y", resulting in "y" to the -3rd power. The answer would be "x" divided by "y" to the 3rd power, turning the negative exponent into a fraction.

When Bases Are Not the Same

When bases are not the same, you cannot divide the exponents.

About the Author

Shannon Hill has served as a writing liaison for educators for the past seven years. Her duties have ranged from working directly with K-5 students on writing strategies to training educators to be effective in writing instruction. She currently serves as both technology coordinator and writing cluster leader.