How to Divide Exponents

Co-authored by David Jia

Last Updated: March 9, 2023 References

Dividing expressions that have exponents is easier than it looks. As long as you're working with the same base, all you have to do is subtract the values of the exponents from each other and keep the base the same. If you're struggling with this, jump down to Step 1 for easy instructions on getting through the process.

Part 1

Part 1 of 2:

Understanding the Basics

1
Write down the problem. The most simple version of this problem will be in the form of m^a ÷ m^b. In this case, you're working with the problem m⁸ ÷ m². Write it down.
2
Subtract the second exponent from the first. The second exponent is 2 and the first is 8. So, you can rewrite the problem as m^8-2.^{[1]
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3
State your final answer. Since 8 - 2 = 6, your final answer is m⁶. It's that simple. If you're not working with a variable and have an actual number in the base, such as 2, then you would have to do the math (2⁶ = 64) to finish solving the problem.^{[2]
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Part 2

Part 2 of 2:

Going the Extra Mile

1
Make sure that each expression has the same base. If you're working with different bases, then you cannot divide the exponents. Here's what you need to know:
- If you're working with a problem with variables, such as m⁶ ÷ x⁴, then there's nothing more you can do to simplify it.
- However, if the bases are numbers and not variables, you may be able to manipulate them so you end up with the same base. For example, in the problem, 2³ ÷ 4¹, you first have to make both bases be "2." All you do is rewrite 4 as 2² and do the math: 2³ ÷ 2² = 2¹, or 2.
  - You can only do this, however, if you can turn the larger base into an expression of squared numbers to make it have the same base as the first.
2
Divide expressions with multiple variables. If you have an expression with multiple variables, then you just have to divide the exponents from each identical base to get your final answer. Here's how you do it:^{[3]
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- x⁶y³z² ÷ x⁴y³z =
- x^6-4y^3-3z^2-1 =
- x²z
3
Divide expressions with coefficients. As long as you're working with the same base, it's no problem if each expression has a different coefficient. Just divide the exponents as you normally would and divide the first coefficient by the second. Here's how:^{[4]
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- 6x⁴ ÷ 3x² =
- 6/3x^4-2 =
- 2x²
4
Divide expressions with negative exponents. To divide expressions with negative exponents, all you have to do is move the base to the other side of the fraction line. So, if you have 3^-4 in the numerator of a fraction, you'll have to move it to the denominator. Here are two examples:^{[5]
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- Example 1:
  - x^-3/x^-7 =
  - x⁷/x³ =
  - x^7-3 =
  - x⁴
- Example 2:
  - 3x^-2y/xy =
  - 3y/(x² * xy) =
  - 3y/x³y =
  - 3/x³

Community Q&A

Question

How can I divide a by b, if both have exponent c?

Donagan

Top Answerer

It would be the fraction (a/b) raised to the power of c.
Question

How do I divide a positive number with a positive exponent by a positive number with a negative exponent?

Community Answer

First of all, the two positive numbers (the bases) have to be the same. If they are, you subtract the exponent in the denominator from the exponent in the numerator. If the denominator's exponent is negative, you treat it as if it were positive and add it to the numerator's exponent. Thus, x^3 ÷ x^(-1) = x^4.
Question

What is 400 divided by 10^3?

Community Answer

400 / 10³ = 400 / 1000 = 0.4.