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Power of a Power Calculator

Enter a base and two exponents to simplify (a^m)^n.

How it is calculated

The tool applies the rule (am)n = am·n: it multiplies the two exponents, m·n, and raises the base a to that single exponent. It then shows the simplified form, the numeric value, and the exponents used. Values of 1,000,000,000 or more, or below 0.0001, are shown in scientific notation, such as 1.099512 × 10¹²; other values are rounded to 6 decimals. The rule only holds for a negative base when both m and n are whole numbers, so a negative base with a fractional m or n shows no result. A zero base needs m above zero and a non-negative combined exponent, and any result that is not a finite number is also not shown.

Worked example

Take base a = 2, inner exponent m = 3 and outer exponent n = 4. Multiply the exponents: 3 × 4 = 12, so (23)4 = 212. As a check, 23 = 8 and 84 = 8 × 8 × 8 × 8 = 4,096, the same as 212. The tool shows 2^(12) = 4,096.

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Common mistakes

  • Adding the exponents instead of multiplying them. Adding is for am × an; for a power of a power, (23)4 is 212, not 27.
  • Using a negative base with fractional exponents. For example, ((−2)2)0.5 is 2, but the shortcut would give (−2)1 = −2. The tool therefore gives no result when the base is negative and m or n is not a whole number.
  • Confusing (am)n with a(mn). The second is a tower, for example 2(34) = 281, which this tool does not compute.
  • Missing that the minus sign can fall on the base or the exponent. (−3)2 is 9, while −32 is −9; enter the base with its sign.
  • Using a zero base with a zero or negative inner exponent, or a negative combined exponent. 0−1 is undefined, so these entries show no result.

About this calculator

The power-of-a-power rule is one of the core exponent laws in algebra: (a^m)^n = a^(m·n). Raising a power to another power doesn’t require expanding anything — you simply multiply the two exponents together and keep the same base, which is a huge shortcut compared to writing out repeated multiplication.

This rule underpins simplifying algebraic expressions, scientific notation, and polynomial work throughout high school and college math, and it’s essential in fields like computer science (analyzing complexity expressed as exponents) and physics (unit conversions with powers of ten). With a base and two exponents, it multiplies the exponents into one, simplifies the expression and gives the numeric value, switching to scientific notation once it gets very large.

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Comments (1)

  • Editorial team

    The rule only collapses two exponents on a stack when parentheses are explicit. (2^3)^4 is 2^12 = 4,096, but 2^(3^4) is 2^81, which the tool displays as 2.417852 × 10²⁴ when you enter m = 81 and n = 1. A tower is read from the top down, so the two readings are different numbers. Have you hit an example where the rule seems to break? Post the numbers.

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