Square Root Calculator
Find the square root, square and cube root of any number instantly.
Three results from one number
Enter a number and the calculator shows its square root (√n), its square (n × n) and its cube root (∛n), each rounded to two decimal places. The default, 144, gives a square root of 12, a square of 20,736 and a cube root of 5.24.
The square root shown is the positive one. Both 12 and −12 multiply by themselves to 144, so if you're solving an equation such as x² = 144, remember the second answer, x = −12.
What happens with negative numbers
| Input | Square root | Square | Cube root |
|---|---|---|---|
| −16 | — | 256 | −2.52 |
| −27 | — | 729 | −3 |
No real number squares to a negative, so the square root shows a dash. In algebra the answer is imaginary, √−16 = 4i, which this calculator doesn't display. Cube roots are fine with negatives, because a negative number cubed stays negative: −3 × −3 × −3 = −27.
Rounded roots, and simplifying them by hand
Most square roots never end, so the two-decimal result is an approximation. √2 shows as 1.41, but 1.41² is 1.9881, not 2. If a later step depends on the root, carry the exact form or more digits, not the rounded display.
Textbooks often want simplified radicals rather than decimals. To simplify √72, find its largest perfect-square factor, 36, and take its root outside: √72 = √36 × √2 = 6√2. The calculator shows the decimal value, 8.49, which you can use to check your simplified answer.
To estimate a root without a calculator, guess, then average your guess with n ÷ guess. For √72, starting at 8:
- (8 + 72 ÷ 8) ÷ 2 = 8.5
- (8.5 + 72 ÷ 8.5) ÷ 2 = 8.4853
- One more round gives 8.48528, which matches the true root in every digit shown.
Everyday uses
- Side of a square from its area: a square patio of 200 sq ft is √200 = 14.14 ft on each side.
- A diagonal from two sides: by Pythagoras, a 12 ft × 16 ft room has a corner-to-corner distance of √(12² + 16²) = √400 = 20 ft. Enter 400 to get it.
- Side of a cube from its volume: a cube holding 64 cubic inches is ∛64 = 4 in along each edge. That's the cube-root line for 64.
Frequently asked questions
Why is the square root of 0.25 bigger than 0.25?+
For any number between 0 and 1, the root is larger than the number itself, because multiplying two numbers below 1 makes something smaller. √0.25 = 0.5, and 0.5 × 0.5 = 0.25. The square line shows 0.06 for 0.25, which is 0.0625 rounded.
How can I tell if a number is a perfect square?+
Its square root is a whole number: 1, 4, 9, 16 and so on up to 225 for 15 × 15. A quick filter is the last digit. Perfect squares never end in 2, 3, 7 or 8, so 7,743 can't be one.
How do I find a fourth root?+
Take the square root twice. For 81, the square root is 9, then enter 9 to get 3, so the fourth root of 81 is 3. Rounding builds up with each step, so for non-perfect powers treat the second result as an approximation.
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<iframe src="https://utilityapp.lat/embed/math/square-root-calculator" title="Square Root Calculator" width="100%" height="660" style="border:0;max-width:760px" loading="lazy"></iframe> <p style="font-size:13px"><a href="https://utilityapp.lat/math/square-root-calculator">Square Root Calculator</a> by UtilityApp</p>