Square Root Calculator

You're checking a hand-calculation and want to confirm √847 to a few decimals, or you need the cube root of 2,197 (it's 13). Type the number and the root index — 2 for square, 3 for cube, anything else for nth root — and the answer comes back.

Negative numbers under even roots aren't real and the calculator says so explicitly rather than pretending. Negative numbers under odd roots are fine: ∛-27 = -3.

For irrational results, the output shows a configurable number of decimal places. For perfect roots (√144 = 12, ∛8 = 2), it shows the exact integer answer, no trailing zeros.

A square root of x is a number y such that y × y = x. So √144 = 12 because 12 × 12 = 144. Every positive number has two square roots — a positive and a negative — but conventionally, 'the' square root means the positive one. The cube root is the analogous operation for three: ∛8 = 2 because 2 × 2 × 2 = 8. The general nth root: the nth root of x is the value that, raised to the n-th power, gives x.

For perfect roots — where the input has a whole-number root — the answer is exact: √225 = 15, ∛125 = 5, √729 = 27. For non-perfect roots, the result is irrational and the calculator returns a decimal approximation. √2 ≈ 1.41421356, with infinitely many non-repeating digits. The discovery that √2 is irrational famously upset the ancient Greek Pythagoreans, who had assumed all quantities were ratios of whole numbers. They were wrong, and the proof is short enough to fit in a paragraph.

A worked example for diagonal calculations: a TV is advertised as '55-inch' (the diagonal). What does that imply about width and height for a 16:9 aspect ratio? The diagonal forms a right triangle with the width and height. If we set the height as h, the width is 16h/9, and by the Pythagorean theorem: (16h/9)² + h² = 55². Solving: h² × (256/81 + 1) = 3025, so h² = 3025 × 81/337 ≈ 727.4, h ≈ 27 inches. The width is then about 48 inches. The calculator handles the square root step; the rest is algebra you set up around it.

Some honest limitations: complex roots (the square root of a negative number) aren't real and produce imaginary numbers (involving i = √−1). Most users don't want imaginary outputs from a square-root calculator, so the calculator returns an error rather than venturing into complex arithmetic. For odd roots — cube root, fifth root — negative inputs work fine: ∛−27 = −3, because (−3)³ = −27. The calculator handles those without complaint.

For very high-precision work — physical constants, scientific computing — JavaScript's double-precision (about 15–17 significant digits) is the practical limit. If you need 50 digits of √2, you need a library that supports arbitrary precision. For everyday math, homework, engineering, and finance, double precision is overkill; you'll round the answer to two or three decimals anyway.