How to use the Square Root Calculator
- Enter any positive number.
- Read the square root to eight decimal places.
- Check the perfect-square flag if you are factorising.
- Use the cube root row for volume-to-length problems.
How the calculation works
The square root of x is the non-negative value that, multiplied by itself, gives x. Every positive number has two real square roots, one positive and one negative, but the radical symbol conventionally denotes the positive one — which is why solving x² = 9 gives x = ±3 while √9 alone is just 3. Negative inputs have no real square root; they require imaginary numbers.
A perfect square has an integer root: 1, 4, 9, 16, 25 and so on. Recognising them speeds up factorisation and simplification of radicals — √72 becomes √(36 × 2) = 6√2 once you spot the perfect-square factor. Cube roots, by contrast, are defined for negative numbers too, since a negative multiplied three times stays negative.
Under the hood, the browser computes roots by hardware-accelerated methods equivalent to Newton–Raphson iteration, which converges quadratically — each step roughly doubles the number of correct digits.
√x = x^(1/2) ; ∛x = x^(1/3) ; Newton step: xₙ₊₁ = (xₙ + a/xₙ) / 2Source: Standard definitions of real roots; Newton–Raphson method for root finding.
Worked example
Finding the side length of a square patio with an area of 42 m².
- Enter 42.
- Square root: 6.4807407.
- Not a perfect square, so the side is irrational.
About 6.48 m per side — order 6.5 m slabs and trim, rather than assuming a whole number.
Frequently asked questions
Why do I get no answer for a negative number?+
Negative values have no real square root. Their roots are imaginary, written as i√|x|.
Does √x mean both roots?+
No. The radical denotes the principal (positive) root. Equations of the form x² = a have two solutions, ±√a.
How do I simplify a radical?+
Pull out the largest perfect-square factor: √50 = √(25 × 2) = 5√2.
Can I get roots other than square and cube?+
Yes — use the exponent calculator and raise to the reciprocal power, such as x^(1/5) for a fifth root.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.