Calculate with Product of Powers
Your result will appear here.
What is a product of powers calculator?
A product of powers calculator applies the same-base product rule aᵐ × aⁿ = aᵐ⁺ⁿ. Expanding both powers places m equal factors beside n more equal factors. Together they form m + n copies of the same base.
Important terms
- Common base
- The identical repeated factor a in both powers.
- Factor
- A quantity multiplied by another quantity.
- Exponent sum
- The combined exponent m + n.
- Product rule
- The rule that adds exponents when same-base powers are multiplied.
When this method is useful
- Combining repeated same-base factors.
- Simplifying monomials and exponential expressions.
- Checking growth calculations with a common scale factor.
- Rewriting negative-exponent products as one reciprocal power.
Product of Powers Calculator
Multiplying powers with the same base joins all equal factors
Formula and variables
- a
- the common base
- m
- the first whole-number exponent
- n
- the second whole-number exponent
- m+n
- the combined exponent
Expansion reveals m factors of a followed by n more factors of a, for a total of m + n.
Domain note: The bases must be identical; this calculator accepts one shared base. Exponents must be whole numbers, and their sum must remain between −1,000 and 1,000.
How the exponent rule works
How to use this calculator
- 1
Enter the shared base.
- 2
Enter the first and second exponents.
- 3
Confirm that the original powers use the same base.
- 4
Add the exponents and keep the base unchanged.
- 5
Evaluate the combined power and inspect exact or reciprocal form.
Step-by-step examples
Example 1
Problem: Evaluate 3² × 3⁴.
Substitution: 3^(2 + 4) = 3⁶
Result: 729
Two factors of 3 plus four factors of 3 make six factors.
Example 2
Problem: Simplify 10³ × 10⁻¹.
Substitution: 10^(3 − 1) = 10²
Result: 100
The negative exponent removes one net factor of 10.
Example 3
Problem: Evaluate (−2)³ × (−2)².
Substitution: (−2)^(3 + 2) = (−2)⁵
Result: −32
The odd combined exponent keeps the negative sign.
Worked exponent example
Rules, edge cases, and related ideas
Same base required
Exponent addition counts copies of one repeated factor only.
Factor counting
The sum m + n equals the number of factors after expansion.
Inverse cancellation
Opposite exponents add to zero, so aⁿa⁻ⁿ = 1 for nonzero a.
Edge cases and limitations
- A zero exponent contributes a multiplicative factor of 1 for a nonzero base.
- Opposite exponents combine to zero.
- A zero base requires both original exponents to be positive.
- Different bases cannot be combined by this rule.
How this differs from a related concept
Product of powers adds exponents for multiplication. A power of a power multiplies exponents, while a quotient of powers subtracts them.
Understand the result
A positive exponent sum gives an ordinary positive power count, zero gives 1 for a nonzero base, and a negative sum produces a reciprocal power.
Common mistakes
- Multiplying the exponents instead of adding them.
- Applying the rule to different bases.
- Adding the bases as well as the exponents.
- Making a negative exponent result negative.
- Simplifying an undefined zero power.
Frequently asked questions
Why are the exponents added?
Expansion joins the factor counts: m copies of a plus n copies of a.
Can the exponents be negative?
Yes. Their signed sum becomes the simplified exponent.
Can the bases be different?
No. Different bases do not share one repeated factor, so this rule does not combine them.
What happens when the exponents are opposites?
They sum to zero and the product is 1 for a nonzero base.
Do I multiply the numerical power values first?
You can, but combining exponents first is usually faster and clearer.