What This Calculator Does

A square root answers "what number, multiplied by itself, gives me this?" — trivial for a perfect square like 144, much less trivial for something like 50, where the answer is an endless decimal. Enter a number and this tool returns its root as a decimal, computed to full floating-point precision rather than left as an unsimplified radical. The optional Root Degree field extends this beyond square roots: leave it at its default of 2 for a square root, or set it to 3, 4, or any other positive whole number for a cube root, 4th root, or higher.

How to Use It

Enter a Number, optionally change the Root Degree from its default of 2, and the result reads back as a sentence: "The square root of [your number] is [result]" (or "cube root" for degree 3, "4th root" and so on for higher degrees). Whole-number results display as plain integers (144 shows a result of 12, not 12.00000), while irrational results are shown to 5 decimal places. Negative numbers are accepted for odd root degrees (3, 5, 7, ...), since an odd root of a negative number is a real, well-defined value — but even root degrees (2, 4, 6, ...) still reject negative input, since there's no real-number result for those; you'd need imaginary numbers, which this tool doesn't handle.

The Formula

For the default square root (degree 2), the calculator finds the non-negative number that, squared, returns your input, using full double-precision floating-point arithmetic. Results are always shown as decimals, never simplified into radical form (√50 is shown as 7.07107, not 5√2).

For any other root degree n, the general formula is nth root = x^(1/n). Cube roots (n=3) use a dedicated cube-root routine rather than raising to the power of 1/3 directly, specifically so that negative inputs still work correctly — the cube root of -8 is a perfectly real -2, but naively computing a negative number raised to a fractional power produces an invalid result in standard floating-point math. That same reasoning applies to every odd degree; even degrees (4, 6, ...) have no real result for a negative input and are rejected with a clear error instead.

A Worked Example

√144: since 12 × 12 = 144 exactly, the result is a clean 12, shown as an integer with no decimal places. Compare that with an input that isn't a perfect square, like √2: there's no whole number that squares to exactly 2, so the calculator returns the full irrational value, 1.41421 (rounded to 5 decimal places from the underlying 1.4142135623730951).

For a non-square example, set Root Degree to 3 and enter -8: since -2 × -2 × -2 = -8, the cube root of -8 is -2, a real result the calculator returns directly. Try the same -8 with Root Degree set to 4 instead, and you'll get a clear error rather than a number, since no real number raised to the 4th power can produce a negative result.

How to Calculate a Square Root Manually

For a perfect square, it's just recall: think of which whole number multiplied by itself gives your number (12 × 12 = 144, so √144 = 12). For anything else, the practical hand method is trial and refinement: guess a number, square it, and adjust up or down based on whether you overshot or undershot. To estimate √50, note that 7² = 49 and 8² = 64, so the answer is just above 7 — trying 7.1² = 50.41 shows it's a little too high, so the true value sits close to 7.07, which matches what a calculator returns (7.07107). The same trial-and-refinement approach works for a cube root by cubing your guess instead of squaring it: for the cube root of 27, 3 × 3 × 3 = 27 exactly, so the answer is a clean 3.