Square Root Calculator
The square root of a number is the value that, when multiplied by itself, gives you the original number. This calculator handles square roots, cube roots, and any nth root.
Last updated: August 2026 · Written and verified by Akash Pandey
Calculate a root
√x
This runs the same calculation engine as the main scientific calculator, entirely in your browser. Nothing you type is sent anywhere.
What is a square root?
The square root of a number x is the value that, when multiplied by itself, gives x. Written as √x. For example, √9 = 3 because 3 × 3 = 9.
√x = y where y × y = x
Why a square root reverses squaring
Squaring and taking a square root are inverse operations, the same relationship as multiplication and division: squaring asks "what do I get if I multiply y by itself?", and the square root asks the question backwards - "which y produced this?". That backwards direction is why square roots are harder to compute by hand than squares: multiplying is direct, but finding y from y² generally means searching or estimating rather than a single calculation, which is exactly why non-perfect-square roots go on forever without repeating (they are irrational).
Estimating a root that is not a perfect square
You do not need a calculator to get close. Find the two perfect squares your number sits between, then judge how far between them it sits.
Estimate √50 without a calculator
√50
- Given: 50 is not a perfect square (49 = 7² and 64 = 8² are the neighbours).
- Method: 50 sits just above 49, so the root sits just above 7.
- Calculation:
50is 1 unit past 49 out of the 15-unit gap to 64, so add roughly1/15 ≈ 0.07to 7. - Interpretation: the estimate, about 7.07, is very close to the true value √50 ≈ 7.071.
= ≈ 7.07 (true value ≈ 7.0711)
Perfect squares reference
| n | n² | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 12 | 144 | 12 |
| 15 | 225 | 15 |
| 20 | 400 | 20 |
Cube roots and nth roots
A cube root asks: what number, when multiplied by itself three times, gives the original? Written as ∛x. More generally, the nth root asks what number, raised to the power n, gives x.
ⁿ√x = y where yⁿ = x
Worked examples
A square garden has an area of 144 m² - how long is each side?
√144
- Given: area = 144 m², a square, so all four sides are equal.
- Method: side length is the square root of the area, since area = side².
- Calculation:
12 × 12 = 144, so√144 = 12. - Interpretation: each side of the garden is 12 metres.
= 12 m
What volume gives a cube with a 3 cm edge, worked backwards?
∛27
- Given: a cube of volume 27 cm³ - we want its edge length.
- Method: edge length is the cube root of the volume, since volume = edge³.
- Calculation:
3 × 3 × 3 = 27, so∛27 = 3. - Interpretation: the cube has 3 cm edges.
= 3 cm
Real-world uses
- Right triangles - the Pythagorean theorem
c = √(a² + b²)finds a diagonal or hypotenuse from two known sides, used everywhere from carpentry to navigation. - Falling objects - the time to fall a height h is
t = √(2h/g), so doubling the drop height does not double the fall time; it multiplies it by √2 ≈ 1.41. - Screen sizes - a TV's diagonal size is
√(width² + height²), which is why a 16:9 "55-inch" screen and a 4:3 "55-inch" screen have different widths. - Statistics - standard deviation is defined as a square root (see the statistics calculator) specifically to undo the squaring used when averaging deviations.
Frequently Asked Questions
What is the square root of a negative number?
In the real number system, the square root of a negative number is not defined. To compute it, you need complex numbers. This calculator operates within the real number system.
How do I simplify a square root?
Factor the number into a perfect square times a remainder. For example, √72 = √(36 × 2) = 6√2.
What is the difference between the principal square root and the negative square root?
The principal square root (√x) is always the non-negative value. Every positive number has two square roots: one positive and one negative. √9 = 3, but -3 × -3 also equals 9.
Are square roots of prime numbers always irrational?
Yes. The square root of any prime number is irrational — it cannot be expressed as a fraction. √2 ≈ 1.41421… goes on forever without repeating.
What is the square root of 0?
The square root of 0 is 0, because 0 × 0 = 0. It is the only number whose square root equals itself.
How do I estimate a square root without a calculator?
Find the two perfect squares the number sits between, then judge the fraction of the way across. For √50: it sits between 49 (7²) and 64 (8²), one unit past 49 out of a 15-unit gap, so the estimate is about 7 + 1/15 ≈ 7.07 - close to the true 7.0711.
Why does this calculator accept ∛(−27) but reject √(−4)?
Odd roots and even roots behave differently for negative numbers. A negative number raised to an odd power stays negative, so (−3)³ = −27 means ∛(−27) = −3 is a perfectly real answer. A negative number raised to an even power is always positive, so no real number squared can give a negative result - that is why √(−4) is rejected rather than answered.
How do I calculate a cube root?
A cube root asks what number, multiplied by itself three times, gives the original. For example, ∛27 = 3 because 3 × 3 × 3 = 27. The calculator above supports cube roots directly.