Power of a Power Calculator

Enter a base and two exponents to compute (a^m)^n using the power of a power rule a^(m×n), with a step-by-step check of the inner and outer exponentiation.

Quick Facts

Power of a power rule
(a^m)^n = a^(m×n)
Multiply the exponents together; the base stays the same.
Product of powers rule
a^m × a^n = a^(m+n)
A different rule — add exponents when multiplying same-base powers, don't confuse it with power of a power.
Negative base caveat
Needs integer exponents
If a < 0, both m and n must be whole numbers for the result to stay a real number.

Your Results

Calculated
(a^m)^n Result
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Final value of the power of a power
Combined Exponent (m × n)
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The two exponents multiplied together
Step 1: a^m
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Inner power evaluated first
Step 2: (a^m)^n check
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Inner result raised to n — should match the result above

Ready

Enter a base and two exponents, then press Calculate.

How the Power of a Power Rule Works

The power of a power rule is one of the core exponent rules: when a power is itself raised to another power, you multiply the exponents and keep the base unchanged. In symbols, (am)n = am×n. This calculator takes a base a and two exponents m and n, evaluates the inner power am, then raises that result to the n-th power, and cross-checks it against the direct calculation am×n.

Formula and derivation

The rule follows directly from what exponents mean. Raising am to the n-th power means multiplying am by itself n times: am × am × ... × am (n factors). Each factor contributes m copies of a to the product, so the total number of a's multiplied together is m × n. That is exactly the definition of am×n. For example, (23)2 = 82 = 64, and directly, 23×2 = 26 = 64 — the same answer either way.

Domain notes: negative and fractional exponents

For a positive base (a > 0), the rule am×n holds for any real exponents m and n, including negative numbers (a-k = 1/ak) and fractions (a1/k is the k-th root of a). For a negative base, the inner step am is only guaranteed to be a real number when m is an integer; a non-integer exponent on a negative base produces a complex number, so this calculator requires both m and n to be whole numbers whenever the base is negative. A base of 0 with a negative combined exponent is undefined (it would require dividing by zero), so that combination is also rejected.

Common sources of error

  • Confusing power of a power with product of powers: (am)n = am×n (multiply exponents) is a different rule from am × an = am+n (add exponents) — mixing them up is the most common mistake.
  • Sign errors with negative bases: (-2)2 = 4, but ((-2)2)3 = 43 = 64, while a non-integer exponent applied directly to -2 does not give a real number.
  • Rounding intermediate steps: for fractional exponents, avoid rounding am before applying the outer exponent — carry full precision through to the final step to avoid compounding error.

Checking your result

This calculator shows both a direct route (computing am×n in one step) and a two-step route (computing am, then raising that to the n-th power). If the two routes disagree by more than a tiny floating-point rounding difference, double check that the exponents and base were entered as intended — for very large exponents, results can also overflow to infinity, which is expected behavior rather than an error.

Frequently Asked Questions

What is the power of a power rule?
The power of a power rule states that (am)n = am×n: when you raise a power to another power, you multiply the exponents and keep the base the same. For example, (23)2 = 23×2 = 26 = 64.
Why does (a^m)^n equal a^(m×n)?
Raising am to the n-th power means multiplying am by itself n times: am × am × ... × am (n times). Each factor contributes m copies of a, giving m×n total copies of a multiplied together, which is am×n.
Does the power of a power rule work with negative or fractional exponents?
Yes, for a positive base the rule am×n holds for any real exponents, including negative and fractional ones. If the base is negative, both exponents must be integers for the result to be a real number; otherwise the intermediate step can produce a complex number.
How is the power of a power rule different from the product of powers rule?
Power of a power, (am)n = am×n, multiplies the exponents. Product of powers, am × an = am+n, adds the exponents. They apply to different situations: one nested exponent, versus two separate factors with the same base.

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