How this instrument works
Raising a power to a power stacks two exponentiations instead of one: (baseᵐ)ⁿ first raises the starting number to the mth power, then raises that whole result to the nth power again. Expand it and the reason the exponents multiply becomes visible rather than assumed: raising to the mth power and then the nth power means n groups of m copies of the original value sitting side by side, and n groups of m copies is mn copies total. The identity (baseᵐ)ⁿ = base^(mn) is just that count, written compactly.
This is easy to confuse with the neighboring law for combining two separate powers of the same number, baseᵐ × baseⁿ = base^(m+n), where the exponents add because the two stacks sit end to end rather than one stack repeated. Power of a power nests one exponentiation inside another instead of placing two side by side, and nesting compounds by multiplication the way sequential percentage changes compound, not by addition the way a single combined stack does. Adding when the number was actually raised twice is the single most common slip students make with exponents.
A one-line proof makes the rule airtight rather than merely plausible: take logarithms of both sides, and n·log(baseᵐ) equals n·m·log(base) by the same power rule applied to the logarithm itself, which is exactly log(base^(mn)) — so the two exponents were always going to land on the same product. Because multiplication doesn't care about order, (baseᵐ)ⁿ and (baseⁿ)ᵐ compute the identical number: swap which exponent sits inside and which sits outside and nothing changes. And at n = 0 the outer power collapses everything to 1 no matter what m did first, since any nonzero result raised to the zeroth power returns 1.
- Enter the number you're raising to a power into the Base field — any positive value works, including fractions and decimals.
- Enter the first exponent, the one applied directly to the base, into the Inner exponent, m field.
- Enter the second exponent, the one applied to that whole inner result, into the Outer exponent, n field.
- Read (baseᵐ)ⁿ: the sheet multiplies m and n internally and reports base raised to that single combined exponent.
- Change either exponent and (baseᵐ)ⁿ updates immediately, so you can compare, say, (2³)⁴ against (2⁴)³ and confirm they match.
Worked example — (2³)⁴ multiplies to 2¹²
Set Base to 2, Inner exponent, m to 3, and Outer exponent, n to 4. The inner step gives 2³ = 8; the outer step then raises that 8 to the 4th power, 8⁴ = 4,096. The shortcut skips the intermediate 8 entirely: multiply the exponents first, 3 × 4 = 12, then compute 2¹² = 4,096 — the same answer, reached without ever forming 8⁴ by hand. (baseᵐ)ⁿ reports 4096.0 directly.
Change the starting number to 10, m to 2, and n to 3, and the same mechanism gives (10²)³ = 10⁶ = 1,000,000 — six zeros, because the exponents multiplied to 2×3=6 rather than adding to 2+3=5, the wrong total a habit borrowed from the product rule would produce. Drop n to 0 with the number 3 and m 2 restored, and (3²)⁰ returns exactly 1, since any nonzero quantity raised to the zeroth power is 1 regardless of what the inner step computed first.
Questions
Why do the exponents multiply instead of add in power of a power?
Because (baseᵐ)ⁿ means baseᵐ multiplied by itself n times — n side-by-side copies of a stack that already holds m copies of the original number, for mn copies total. Addition belongs to a different rule, baseᵐ × baseⁿ = base^(m+n), which combines two separate powers rather than nesting one inside another.
How is power of a power different from multiplying two separate powers?
Multiplying two separate powers of the same number, baseᵐ × baseⁿ, adds the exponents because the two stacks sit end to end: base^(m+n). Power of a power nests one exponentiation inside another instead, and nesting multiplies: (baseᵐ)ⁿ = base^(mn). Mixing the two up, adding when the value was actually raised twice, is the most common mistake with this identity.
Does the order of the two exponents matter?
No. Because m×n and n×m are the same product, (baseᵐ)ⁿ and (baseⁿ)ᵐ always compute the identical number — for example (2³)⁴ and (2⁴)³ both equal 2¹² = 4,096. Swapping which exponent is applied first never changes the final result.
What happens when the outer exponent, n, is 0?
The result is 1, regardless of what the number or the inner exponent, m, computed. Any nonzero value raised to the zeroth power equals 1, and that rule applies to the whole inner result (baseᵐ) just as it applies to the plain number alone — (3²)⁰, for instance, returns exactly 1 even though 3² itself is 9.
Can I prove (baseᵐ)ⁿ = base^(mn) without just trusting the pattern?
Yes — take logarithms of both sides. The log of (baseᵐ)ⁿ equals n·log(baseᵐ), which equals n·m·log(base) by the same exponent rule applied one level down; that is exactly the log of base^(mn). Since a logarithm returns equal outputs only for equal inputs, the two original expressions must be equal too.
Where does this identity actually get used outside a textbook?
Anywhere a rate compounds over nested periods. Interest compounded quarterly for several years, for instance, raises a per-quarter growth factor to a power for one year, then raises that annual factor to a power again for the number of years — the two exponents multiply exactly the way this identity predicts. Repeated-doubling growth and decay models share the same nested structure.