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.