Calculate with Power of a Product
Your result will appear here.
What is a power of a product calculator?
The power-of-a-product rule says that an integer power of a product can be distributed to each factor: (ab)ⁿ = aⁿbⁿ. The product ab is repeated as a factor n times. For negative integer n, the product must be nonzero and the expression means the reciprocal of the corresponding positive power.
Important terms
- Product
- The result of multiplying factors; here the product is ab.
- Base
- The quantity being raised to a power. In (ab)ⁿ, the grouped product ab is the base.
- Exponent
- The integer n that counts repeated factors; a negative integer indicates a reciprocal.
- Power rule
- The identity (ab)ⁿ = aⁿbⁿ, which moves the exponent to every factor inside parentheses.
When this method is useful
- Expanding powers of monomials such as (3xy)⁴.
- Simplifying scientific-notation and scale-factor calculations.
- Checking algebraic transformations before evaluating numbers.
- Separating a product into factors that are easier to calculate independently.
Power of a Product Calculator
An exponent reaches every factor inside the parentheses
Formula and variables
- a, b
- the two real factors
- n
- an integer exponent
- (ab)ⁿ
- the product raised to the exponent
For positive n, expand (ab)ⁿ as n copies of ab, then regroup all a factors and all b factors. For negative n, use x⁻ⁿ = 1/xⁿ, which requires ab ≠ 0.
Domain note: Finite real factors with up to 120 digits and integer exponents from −1,000 to 1,000 are supported. Zero to a zero or negative exponent is undefined. Results above 5,000 digits use rounded scientific notation.
How the formula works
How to use this calculator
- 1
Enter the first and second real factors.
- 2
Enter an integer exponent; a negative exponent is allowed only when the product is nonzero.
- 3
The calculator forms the product and applies the same exponent to each factor.
- 4
Compare (ab)ⁿ with aⁿ × bⁿ to see the identity.
- 5
Check whether the displayed result is exact or rounded scientific notation.
Step-by-step examples
Example 1
Problem: Evaluate (2 × 3)².
Substitution: (2 × 3)² = 2² × 3² = 4 × 9
Result: 36
Both factors receive the exponent, and the two squares multiply to the same result as 6².
Example 2
Problem: Simplify (−2 × 3)³.
Substitution: (−2)³ × 3³ = −8 × 27
Result: −216
An odd power preserves the negative sign of the negative factor.
Example 3
Problem: Evaluate (0.5 × 4)².
Substitution: 0.5² × 4² = 0.25 × 16
Result: 4
Distributing the exponent can make decimal factors easier to inspect.
Worked example, step by step
Rules, edge cases, and related ideas
Parentheses define the base
In (a × b)ⁿ, the exponent applies to the entire product, so it reaches both factors.
Integer exponent domain
This real-valued calculator supports integer powers, avoiding branch ambiguities for fractional powers of negative factors.
Zero exponent
For a nonzero product, (ab)⁰ = a⁰b⁰ = 1. If a factor makes the product zero, 0⁰ is undefined here.
Edge cases and limitations
- A zero factor is valid for a positive exponent and makes the result zero.
- A zero product cannot use exponent zero or a negative exponent.
- Negative factors are valid for integer exponents; parity controls the sign.
- Negative exponents may produce repeating decimals, so a decimal display can be rounded.
- Very large exact values are represented in scientific notation instead of being printed in full.
How this differs from a related concept
This rule distributes an exponent over multiplication. It differs from the product-of-powers rule aᵐaⁿ = aᵐ⁺ⁿ, which combines two powers that already have the same base.
Understand the result
The distributed expression and grouped product are algebraically identical. A positive exponent follows the usual parity rules; a negative exponent reciprocates the complete product power.
Common mistakes
- Applying the exponent to only one factor.
- Forgetting parentheses and treating −aⁿ as (−a)ⁿ.
- Using the identity with a non-integer exponent without checking real-domain conditions.
- Assuming a negative exponent makes the result negative rather than reciprocal.
Frequently asked questions
Does the exponent apply to both factors?
Yes. Parentheses make ab the base, so (ab)ⁿ = aⁿbⁿ.
Can the factors be negative?
Yes, for integer exponents. Each factor follows the usual parity rules.
Can I use a negative exponent?
Yes, provided ab is nonzero. A negative exponent means take the reciprocal of the positive power.
Why does the rule work?
For positive integer n, write n copies of ab and regroup the factors. Reciprocal definitions extend the identity to negative integers when the product is nonzero.
Does (a+b)ⁿ equal aⁿ+bⁿ?
No. This rule distributes over multiplication, not addition. In general, a sum power has additional cross terms.