Exponent Calculator

Exponents compress repeated multiplication into a tiny superscript, and this calculator works in all three directions: compute a raised to the n, solve for the exponent when you know the base and the result, or solve for the base when you know the exponent and the result. It shows the full expansion for small integer powers, accepts scientific notation, and refuses to fake answers for undefined cases like 0^0.

Educational tool. Results use standard floating-point arithmetic; very large powers may display in scientific notation or as overflow.

The exponent rules that actually matter

You do not need to memorize a wall of laws; five cover nearly everything:

• Negative exponent: a^-n = 1 / a^n. So 2^-2 = 1/4 = 0.25.
• Zero exponent: a^0 = 1 for any non-zero a. (0^0 is undefined; the calculator says so instead of guessing.)
• Fractional exponent: a^(m/n) is the nth root of a^m. 9^0.5 is the square root of 9, which is 3.
• Power of a power: (a^m)^n = a^(m x n). Handy for breaking big exponents into steps.
• Same base multiplication: a^m x a^n = a^(m+n). This is why 2^10 x 2^3 = 2^13 = 8,192.

Worked examples, verified against the tool

Example 1, positive exponent: base 2, exponent 10. The calculator expands it as 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 and returns 1,024.

Example 2, negative exponent: base 2, exponent -2. Shown as 1 / (2 x 2), returning 0.25.

Example 3, solving for the exponent: "2 to what power equals 1,024?" Switch to the "Solve for n" mode, enter base 2 and result 1,024. The tool computes n = log(1024) / log(2) = 10.

Example 4, solving for the base: "What cubed equals 27?" In "Solve for a" mode, enter exponent 3 and result 27. The tool computes the cube root: 3.

Each of these was run through the actual calculator code and the outputs match exactly.

Edge cases the calculator handles honestly

Most exponent tools either crash or silently return a wrong-looking number on tricky inputs. This one explains itself:

• 0^0: reported as undefined (mathematicians genuinely disagree on a convention, so no silent choice is made).
• 0 to a negative power: undefined, because it means dividing by zero.
• Negative base with a fractional exponent: not a real number (for example (-8)^0.5); the calculator says so rather than printing NaN.
• Solving for n with base 1: impossible, since 1 to any power is 1; the tool tells you instead of dividing by log(1) = 0.

Common mistakes

The most frequent mistake is typing a negative base with a fractional exponent and trusting whatever the tool prints; if the result is not a real number, no real answer exists. The second is confusing a^(m/n) order of operations: it is the nth root of (a^m), not something you can split arbitrarily. The third is forgetting that exponentiation is not commutative: 2^3 = 8 but 3^2 = 9, so base and exponent are not interchangeable.

Frequently asked questions

What is an exponent?

An exponent tells you how many times to multiply the base by itself. In 2^3, the base 2 is multiplied by itself 3 times: 2 x 2 x 2 = 8.

What does a negative exponent mean?

Take the reciprocal: a^-n = 1 / a^n. So 2^-2 = 1 / 4 = 0.25.

What is a fractional exponent?

A root combined with a power. a^(1/2) is the square root of a, a^(1/3) the cube root. 9^0.5 = 3.

What is 0^0?

Undefined in standard arithmetic; this calculator reports it as undefined rather than picking a convention silently.

Can I enter numbers in scientific notation?

Yes. Type 4e2 for 400 or 2e-6 for 0.000002; the input fields accept them directly.

How do I find the exponent if I know the base and result?

Use the "Solve for n" mode: n = log(result) / log(base). For example, 2^n = 1024 gives n = 10.

How do I find the base if I know the exponent and result?

Use the "Solve for a" mode, which computes the nth root of the result. a^3 = 27 gives a = 3.

Why does a negative base with a decimal exponent fail?

Because the result is not a real number. Roots of negative numbers with non-integer exponents live in complex numbers, which this calculator does not attempt.