This calculator divides one power by another and shows you the full working behind the answer, not just the final number. Enter a base and exponent for the first power (the numerator) and a base and exponent for the second power (the denominator); both can be positive, negative or decimal values. The tool first checks whether the two bases match. If they do, it applies the quotient of powers rule, subtracting the exponents and keeping the base the same, and displays the rule applied, the exponent subtraction, the simplified form and the final value. If the bases differ, that shortcut does not apply, so the calculator works out each power separately and divides the numerator value by the denominator value directly instead. Either way you get the numerator and its calculated value, the denominator and its calculated value, the overall quotient, and confirmation of whether the bases were the same, along with a plain-English summary of which method was used and how the result was reached. It is useful for checking maths homework, verifying exponent laws by hand, or exploring what happens when subtracting exponents gives a zero or negative result, which produces 1 or a fraction respectively. Very large exponents may display in scientific notation, and results are undefined where the base is zero and the exponent is zero or negative.
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Method verified
Standard exponent (index) laws, quotient of powers rule: a^m ÷ a^n = a^(m-n).
1. First Power (Numerator)
^
2. Second Power (Denominator)
^
Result
Numerator
2^5
= 32
Denominator
2^2
= 4
Quotient
8
2^3
Same Base?
Yes
Determines the method used
Enter two powers above to see the working.
Working (Same Base Method)
Rule applieda^m ÷ a^n = a^(m-n)
Exponent subtraction5 - 2 = 3
Simplified form2^3
Final value8
Working (Direct Calculation)
Numerator value32
Denominator value4
Division result8
Summary:Enter your two powers above.
How Dividing Exponents Works
Dividing exponents means dividing one power by another. A power (or exponent) is a base number multiplied by itself a certain number of times, written as a^n, where a is the base and n is the exponent. When you divide two powers, the method depends on whether the bases are the same or different.
Same Base: The Quotient of Powers Rule
When two powers share the same base, you can simplify the division by subtracting the exponents and keeping the base the same:
am ÷ an = a(m-n)
For example, 25 ÷ 22 = 2(5-2) = 23 = 8. This works because dividing repeated multiplication cancels out matching factors. Writing 25 as 2×2×2×2×2 and 22 as 2×2, two of the factors on top cancel with the two on the bottom, leaving three factors of 2, which is 23.
Zero and Negative Exponents
Result of m - n
Meaning
Example
Positive
Standard power, a(m-n)
25 ÷ 22 = 23 = 8
Zero
Any non-zero base to the power 0 equals 1
53 ÷ 53 = 50 = 1
Negative
a-n = 1 ÷ an, a fraction
22 ÷ 25 = 2-3 = 1/8 = 0.125
Different Bases
If the two powers have different bases, the quotient of powers shortcut does not apply directly. Instead, calculate each power separately, then divide the results as ordinary numbers. For example, 34 ÷ 23 = 81 ÷ 8 = 10.125. There is no single exponent rule that simplifies this further unless the bases can be rewritten with a common factor (for example, 43 ÷ 23 can be rewritten as (4÷2)3 = 23 = 8, using the rule an ÷ bn = (a÷b)n when the exponents match).
Worked Example
Divide 25 by 22. Since both bases are 2, subtract the exponents: 5 - 2 = 3. The answer is 23, which equals 8. Checking directly: 25 = 32 and 22 = 4, and 32 ÷ 4 = 8, confirming the same result.
Sources: Standard index laws (laws of exponents) as taught in the NZ Mathematics and Statistics curriculum. Quotient of powers rule: am ÷ an = a(m-n) for any non-zero base a.
This calculator handles real number bases and integer or decimal exponents. Very large exponents may return results in scientific notation or Infinity due to the limits of floating point arithmetic. For a base of 0, results are undefined where the exponent is zero or negative.
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