Exponent Calculator - Powers, Roots, and a^b

An exponent calculator evaluates powers a^b, nth roots, and - when the power is huge - the scientific form a × 10^k. People search for how to calculate exponents, what 2^10 is, how negative exponents work, and how fractional exponents relate to roots because those four questions are the entire high-school power toolkit.

A positive integer exponent is repeated multiplication: 2^10 means ten factors of 2, which is 1024. That identity is why a kibibyte is 1024 bytes and why 2^n shows up in binary addressing. Negative exponents flip to a reciprocal: 2^(−3) = 1/8. Fractional exponents are roots: 9^(1/2) = 3. The identities are clean on paper and messy on a computer once you hit 0^0, a negative base with a non-integer exponent, or a result larger than about 1 × 10³⁰⁸.

How to calculate exponents by hand is still the integer story: 2^10 is ten twos. Negative exponents and fractional exponents need the log identity this page prints. Roots versus powers is the same identity in the opposite direction, which is why Nth-root is a separate mode with its own even-root error instead of a silent NaN. Privacy is unchanged: the base and exponent never leave the browser.

This free online exponent calculator runs in the browser on IEEE-754 binary64 floats. It does not use eval(). It rejects 0^0 as undefined. It rejects a negative base with a non-integer exponent rather than returning a complex number. Even roots of negatives are rejected in Nth-root mode. If a^b would overflow to Infinity, you get a clear error instead of a hung tab or a silent Inf.

The log identity shown in the result panel is a^b = exp(b × ln|a|), with the sign restored when the exponent is an integer. That is the same identity a log calculator inverts. For rewriting an already-computed value as a × 10ⁿ without taking a new power, use the scientific notation calculator. For mixed trig and memory buttons, a scientific calculator is the broader sibling.

Using this power and root calculator is straightforward:

1. Choose a mode. Power computes base^exponent. Nth root computes the n-index root of x (n a non-zero integer). Scientific computes the same power as Power mode but displays a × 10^k when the magnitude is large.
2. Enter the numbers. Decimals and E-notation such as 2.5e-3 are allowed. For a cube root of a negative, use Nth-root with n = 3, not Power with exponent 1/3.
3. Read the value and the log identity. Steps show ln|a|, the product b × ln|a|, exp of that product, and the restored sign when the exponent is an integer.
4. Copy the result, or clear the fields. The last inputs are saved locally for up to 30 days.

Note: this is IEEE float, not arbitrary precision. 2^53 + 1 is not exact. Values beyond about 1e308 overflow. Values near 1e−308 underflow toward 0. The identity exp(b ln|a|) is pedagogic; the numeric value uses Math.pow so integer powers stay as accurate as the platform allows.

For a > 0 the real logarithm is defined, so every real exponent is fair game. For a < 0 only integer exponents stay inside the reals. Roots are the inverse: the n-th root of x is x^(1/n) when that power exists.

Example - 2^10 (the default):

• 2×2×2×2×2×2×2×2×2×2 = 1024
• ln|2| ≈ 0.693147, 10 × ln2 ≈ 6.93147, exp of that ≈ 1024
• Scientific form: 1.024 × 10^3

Example - negative exponent, 2^(−3):

• a^(−n) = 1 / a^n
• 2^(−3) = 1/8 = 0.125
• The minus on the exponent is a reciprocal, not a negative answer

Example - cube root of −8:

• Nth-root mode, x = −8, n = 3
• Odd integer n, so the real cube root exists: −2
• Power mode with exponent 0.333… is rejected because 1/3 is not an integer in IEEE float. That is intentional, not a bug.

Roots and powers are inverses only when the operations stay real. (−8)^(1/3) is the cube root −2 in real arithmetic, but (−8)^0.3333333333 is a slightly different real-or-complex question because 0.3333333333 is not exactly 1/3. This page refuses to pretend they are the same. Scientific mode does not compute a different function; it only changes how a finite power is printed.

The most common school error is reading 2^3 as 2×3 = 6 instead of 8. Another is thinking a negative exponent makes a negative number: 2^(−3) is +0.125. Order of operations on a^b^c is right-associative in some languages and left in others; this tool only takes one exponent, so you must decide the grouping.

Fractional exponents of negatives are the next trap. (−4) ^(1/2) is not −2; the principal square root of −4 is imaginary. (−8)^(1/3) is real, but typing 1/3 as 0.33 is not the same exponent. Use Nth-root with integer n.

0^0 is left undefined here because analysis, combinatorics, and programming languages disagree. 0^positive is 0. 0^negative is a division by zero. Those three cases are not one rule.

The product rule a^m × a^n = a^(m+n) is why 2^3 × 2^7 = 2^10 = 1024. The quotient rule a^m / a^n = a^(m−n) is why 2^10 / 2^3 = 2^7 = 128. The power rule (a^m)^n = a^(mn) is why (2^5)^2 = 2^10. This calculator computes one power at a time; to check a law, compute both sides separately and compare. Floating-point will agree for small integers and drift for awkward decimals.

a^0 = 1 for any nonzero a. 0^positive is 0. 0^0 is rejected. (ab)^n = a^n b^n still holds for positive a, b. (a/b)^n = a^n / b^n when b ≠ 0. Negative exponents are reciprocals, not sign flips: (1/2)^(−3) = 8. Fractional exponents on a positive base are roots: 81^(1/2) = 9, which matches Nth-root mode with x = 81 and n = 2.

What the laws do not license: a^(b^c) is not (a^b)^c in general. 2^(3^2) = 2^9 = 512, while (2^3)^2 = 64. This tool takes a single exponent, so you must pick the grouping. They also do not license a negative base with a non-integer exponent. And they do not cancel IEEE overflow: 10^400 is not a slightly rounded 10^308, it is outside binary64, and the calculator stops.

JavaScript numbers are IEEE-754 binary64. The largest finite value is about 1.797 × 10^308. Math.pow(10, 309) is Infinity. A page that returned Infinity for 10^1000000 would not hang the event loop, but it would look like a successful calculation. This calculator treats overflow as an error so homework is not graded against Inf. Underflow toward 0 for tiny values such as 10^(−400) can still happen; that is the other end of the same format, and it is a limit, not a second formula.

Integer powers that fit still have a precision limit: integers larger than 2^53 are not all representable. 2^10 = 1024 is exact. 2^100 is about 1.2676506 × 10^30 and Scientific mode will show that coefficient, but it is not the integer with every digit printed. If you need every digit of 2^1000, this is the wrong tool.

The displayed identity a^b = exp(b ln|a|) is how real analysis defines real exponents, and it is how you connect this page to a log calculator. The numeric value still comes from Math.pow so that 2^10 stays 1024 rather than a slightly noisy exp . For rewriting a number you already have as a × 10^n, use the scientific notation calculator; Scientific mode here always computes a power first. People type “how to calculate exponents” and “what is 2^10” into a search box because those are the two checks that tell you a power tool is honest: 2^10 must be 1024, not 20, and overflow of 10^400 must not look like a successful 1e308. Negative exponents, fractional exponents on a positive base, and odd roots of negatives are the rest of the real-number story this page will do. Complex values, symbolic a^(m/n) in lowest terms, and exact 1000-digit integers are the story it will not. Roots versus powers is the remaining vocabulary: a power asks for a^b, a root asks which y satisfies y^n = x. Those two questions share the same log identity and they do not share the same error cases, which is why Nth-root is a separate mode.