Exponent Calculator

A negative exponent is a reciprocal, not a negative answer. b^-n is defined as 1 divided by b^n, so 10^-3 is 1 ÷ 1000 = 0.001 and 2^-1 is 0.5. It falls straight out of the division law: b^m ÷ b^n = b^(m-n), so b^0 ÷ b^n = b^-n, and since b^0 is 1 that is 1 ÷ b^n. The sign of the answer is decided by the base, never by the exponent — 2^-3 is a small positive number, and only a negative base can produce a negative result.

A fractional exponent is a root. b^(1/q) is the q-th root of b, so 9^(1/2) is the square root of 9 and 8^(1/3) is the cube root of 8, which is 2. A general fraction combines the two: b^(p/q) is the q-th root of b raised to the power p, so 8^(2/3) is 2 squared, which is 4. That is why 2^0.5 comes out as 1.414214 — it is the square root of 2. This calculator reads a decimal exponent as the simplest fraction that matches it, so 0.2 is recognised as one fifth and the working shows the root form.

This calculator answers 1, which is the convention algebra and combinatorics use, and the one JavaScript, Python and the IEEE 754 pow function all return. The reasons are structural: the empty product is 1, there is exactly one way to choose nothing from nothing, and the binomial theorem and every power series with a constant term need x^0 to be 1 at x = 0. Analysis usually calls 0^0 indeterminate instead, because the two-variable function x^y has no single limit as x and y both approach zero — 0^y is 0 for every positive y, while x^0 is 1 for every nonzero x, so the value you approach depends on the path you take. Both camps are right about different questions. A calculator lives in the discrete setting, so it answers 1 and says so.

Because floating-point pow is defined to return NaN for a negative base with any non-integer exponent. It has no way to tell the exponent 1/3, which has a perfectly good real cube root of -2, from 0.3333333333333333, which does not: over the complex numbers the principal value of a negative number raised to a fractional power is complex, and returning it would be wrong for a real-valued function. This calculator handles the case explicitly. It matches the exponent to the simplest fraction p/q that agrees with it, and if q is odd it returns the real root — (-8)^(1/3) = -2, (-32)^0.2 = -2 — with the sign taken from whether p is odd. If q is even, as it is for 0.5 or 2.5, there is genuinely no real answer, and it refuses with an explanation instead of returning NaN.

A double-precision number stops at about 1.8 × 10^308, and anything past that becomes Infinity, which is why many calculators show a dash for something like 1000^400. The magnitude here is tracked separately as n × log10(b), which is ordinary arithmetic on numbers that never get large, so 1000^400 is reported as about 1.000 × 10^1200 rather than as an error. Only four significant figures are shown, because only four are known: the leading digits come from the fractional part of the logarithm, and the rest were never computed. The same applies underneath, where results below about 5 × 10^-324 collapse to zero as plain numbers but still have a reportable magnitude.

Because 1.1 cannot be stored exactly in binary — it is a repeating fraction in base 2, exactly as one third is in base 10 — so the stored value is a hair off and squaring it doubles the error. This calculator avoids the problem rather than rounding it away. For a whole-number exponent it computes the power in exact integer arithmetic: 1.1 is 11/10, so 1.1 squared is 121 ÷ 100 = 1.21 exactly. The exact result is only used after it has been checked against the ordinary floating-point answer and the two agree, so a genuine difference is never rounded into agreement.

Three, all shown as checks under the working. The product law says b^m × b^n = b^(m+n), because multiplying m factors by n factors gives m + n factors. The power law says (b^m)^n = b^(mn), because m factors repeated n times is mn factors. And the reciprocal law says b^-n = 1 ÷ b^n. The page evaluates both sides of each of these with your own numbers, so the identities are demonstrated on the result rather than merely quoted at you.