Exponent Calculator

You need to compute 2^32 to size a buffer, or 1.07^15 to project a 7% growth rate over 15 years, and pulling out a calculator app for what should be a single keystroke feels needlessly heavy. Exponentiation is one of the operations where intuition fails fast — the difference between 2^10 and 2^20 is a factor of 1024, not 2.

The calculator handles integer and floating-point bases, positive and negative exponents, and fractional exponents (which is just rooting in disguise). Negative exponents return the reciprocal: 2^-3 is 1/8. Fractional exponents like 0.5 give you the square root. Anything multiplied by a power of zero is one — a property that's mostly trivia until you need to handle x^0 in a script and your edge case isn't crashing.

The core operation: x^n means x multiplied by itself n times for positive integer n. The definition extends in a few directions. For n = 0, the result is defined as 1 (which keeps the laws of exponents consistent). For negative n, x^-n = 1/(x^n). For fractional n = 1/k, x^(1/k) is the kth root of x. For arbitrary real n, the operation is defined via the exponential and natural log: x^n = e^(n × ln(x)), which is how computers actually compute it under the hood. This is also why negative bases with non-integer exponents return non-real results — ln of a negative number isn't defined in the reals.

A worked example: compound interest projection. You have $10,000 invested at 7% annual return. After 15 years: 10,000 × 1.07^15 = 10,000 × 2.7590 = $27,590. After 30 years: 10,000 × 1.07^30 = 10,000 × 7.6123 = $76,123. The Rule of 72 estimates that 72/7 ≈ 10.3 years to double, and indeed 1.07^10 = 1.967, close to 2. Exponential growth is wildly different from linear — people consistently underestimate compound returns over long horizons because the curve flattens until it doesn't.

Where exponentiation breaks down practically: floating-point precision. JavaScript's Number type (used by most browser calculators) is double-precision IEEE 754, which means about 15-17 significant digits. Compute 2^53 and you're at the limit; integers above that lose precision. Very large or very small results overflow to Infinity or underflow to zero respectively (the practical limit is around 1.8e308 for the magnitude). For exact arbitrary-precision arithmetic, use BigInt or a math library — the calculator here uses standard floats and is fine for most practical work but not for cryptographic-scale calculations. Negative bases with fractional exponents return NaN because the result isn't a real number; if you need complex-number support, that's a different tool.