Math
Dividing Exponents Calculator
Divide two powers and see the quotient rule in action.
Same base: a^m ÷ a^n = a^(m-n), so 5^2 ÷ 5^5 = 5^-3 = 1/125. Same exponent: a^n ÷ b^n = (a/b)^n. Bases can be numbers or a letter, and any negative exponent in the answer is rewritten as a fraction.
A number (5, -2, 1/2) or a letter (x).
Try an example
Result
Simplified
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- Original expression
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- Value
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- Rule
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Student quick launch
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Study path
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Two rules for dividing powers
| Situation | Rule | Example |
|---|---|---|
| Same base | a^m ÷ a^n = a^(m-n) | x^5 ÷ x^2 = x^3 |
| Same exponent | a^n ÷ b^n = (a/b)^n | 6^2 ÷ 3^2 = 2^2 = 4 |
| Neither | no shortcut; evaluate each | 2^5 ÷ 3^2 = 32 / 9 |
Why the exponents subtract
x^5 ÷ x^2 = (x·x·x·x·x) / (x·x). Two x's cancel from top and bottom, leaving x·x·x = x^3. Subtracting the The small raised number saying how many times to multiply the base by itself. counts how many factors survive the cancelling.
When the exponent goes negative or zero
- x^2 ÷ x^5 = x^-3 = 1 / x^3: more factors on the bottom leave a The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5..
- x^4 ÷ x^4 = x^0 = 1: everything cancels.
- This is exactly why negative exponents mean reciprocals and a zero exponent means 1.
How to use it
- Choose same The number being raised to a power, or the side a shape's height is measured from. if the The top number of a fraction; it counts how many parts you have. and The bottom number of a fraction; it says how many equal parts make one whole. share a base, or same exponent if they share an exponent.
- Enter the base(s) and exponent(s). A letter base gives a symbolic answer.
- Read the simplified power. If the exponent came out negative, the reciprocal form is shown too.
How to read the answer
Subtracting exponents cancels matching factors on top and bottom. A positive result means more factors were on top; a negative result means more were on the bottom, so the answer is a fraction; zero means they cancelled completely and the answer is 1.
Common mistakes and edge cases
- Dividing the exponents instead of subtracting: x^6 ÷ x^2 = x^4, not x^3.
- Subtracting in the wrong order: it is numerator exponent minus denominator exponent.
- Dropping the negative sign: 5^2 ÷ 5^5 = 5^-3 = 1/125, not 5^3.
- Trying to combine different bases with different exponents; no rule applies.
Worked examples
Negative result
5^2 ÷ 5^5 = 5^-3 = 1/125
Simplified
5^(-3)
Variable base
x^5 ÷ x^2 = x^3
Simplified
x^3
Same exponent
6^2 ÷ 3^2 = 2^2 = 4
Simplified
2^2
Exponents cancel
7^3 ÷ 7^3 = 7^0 = 1
Simplified
7^0
Negative exponents
2^-1 ÷ 2^-4 = 2^3 = 8
Simplified
2^3
Fraction bases
(1/2)^3 ÷ (1/4)^3 = 2^3 = 8
Simplified
2^3
Division by zero
Dividing by 0^n is undefined
Simplified
Error
Frequently asked questions
What is the rule for dividing exponents with the same base?+
Keep the base and subtract the exponents: a^m ÷ a^n = a^(m-n). For example, x^5 ÷ x^2 = x^3.
What if the exponent in the denominator is bigger?+
The result has a negative exponent, which means a reciprocal: 5^2 ÷ 5^5 = 5^-3 = 1/125.
How do you divide exponents with different bases?+
If the exponents match, divide the bases: 6^2 ÷ 3^2 = (6/3)^2 = 4. Otherwise evaluate each power separately.
What is x^4 divided by x^4?+
x^0 = 1, for any nonzero x. Everything cancels.
How do you divide negative exponents?+
Subtract them like any other exponents: 2^-1 ÷ 2^-4 = 2^(-1 - (-4)) = 2^3 = 8.
Why can't the base be zero?+
Because a power of 0 in the denominator is 0, and dividing by 0 is undefined.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026