Apply the power of a power rule instantly — enter a base and two exponents to compute (xᵐ)ⁿ = x^(m·n) with a step-by-step solution.
The Power of a Power Rule
When an exponent expression is raised to another power, the exponents multiply:
(x^m)^n = x^(m·n)
Where:
- x is the base
- m is the first (inner) exponent
- n is the second (outer) exponent
- m·n is the combined exponent
Example: (2^3)^4 = 2^(3·4) = 2^12 = 4,096
How to Apply the Rule
Problem: Compute (3^2)^3.
Solution:
- Apply the rule: (3^2)^3 = 3^(2·3)
- Multiply the exponents: 2 × 3 = 6
- Compute: 3^6 = 729
Common Examples
| Expression | Combined Exponent | Result |
|---|
| (2^2)^3 | 2 × 3 = 6 | 2^6 = 64 |
| (3^2)^2 | 2 × 2 = 4 | 3^4 = 81 |
| (10^3)^2 | 3 × 2 = 6 | 10^6 = 1,000,000 |
| (5^0)^n | 0 × n = 0 | 5^0 = 1 |
| (x^1)^n | 1 × n = n | x^n |
Why Does This Rule Work?
(x^m)^n means multiplying x^m by itself n times:
x^m · x^m · … · x^m (n copies)
When multiplying powers with the same base, the exponents add: x^(m + m + … + m). Since m is added n times, the sum is m·n. Therefore (x^m)^n = x^(m·n).
Sources
Frequently Asked Questions
What is the power of a power rule?
The power of a power rule states that when you raise an exponent expression to another power, you multiply the exponents: (x^m)^n = x^(m·n). For example, (2^3)^4 = 2^(3·4) = 2^12 = 4,096.
Why does the power of a power rule work?
Because (x^m)^n means multiplying x^m by itself n times: x^m · x^m · ... · x^m (n times). When multiplying powers with the same base, you add the exponents, giving x^(m+m+...+m) = x^(m·n).
What happens when one of the exponents is 0?
If either exponent makes the combined product m·n equal to 0, then x^0 = 1 for any non-zero base. For example, (5^3)^0 = 5^(3·0) = 5^0 = 1, and (5^0)^4 = 5^(0·4) = 5^0 = 1.
Can the exponents be negative or decimal?
Yes. Negative exponents follow the reciprocal rule: (2^(-2))^3 = 2^(-6) = 1/64 ≈ 0.0156. Decimal exponents are also supported — for example, (4^0.5)^2 = 4^(0.5·2) = 4^1 = 4.
Can the base be negative?
Yes, when the combined exponent m·n is an integer. For example, ((-2)^3)^2 = (-2)^6 = 64. A negative base with a non-integer combined exponent produces a complex number, which this calculator does not support.