Square Root Calculator
The square root and cube root of a number.
Input sheet
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A square root answers "what number, multiplied by itself, gives this?" It shows up whenever you reverse an area into a side length, solve a Pythagorean problem, or work with standard deviation.
How it works
Finds the value that, multiplied by itself, equals the number.
Every positive number has two square roots, one positive and one negative, but the principal (positive) root is what is usually meant and what this tool reports. The cube root is also shown, which works for negative numbers as well.
Perfect squares (4, 9, 16, 25, 144) have whole-number roots; everything else is irrational, an endless non-repeating decimal that the tool rounds for display.
Square root of n = the value that, multiplied by itself, equals n
Worked examples
The square root of 144. → 12
12 × 12 = 144, so 144 is a perfect square.
The square root of 2. → ≈ 1.4142
1.4142 × 1.4142 ≈ 2; the true value is irrational and never terminates.
Tips & gotchas
- Square roots of negative numbers are not real — they require imaginary numbers, which this tool flags as undefined.
- The cube root, unlike the square root, is defined for negatives: the cube root of −27 is −3.
- To estimate a root by hand, find the two perfect squares it sits between. The square root of 50 is just over 7, since 7² = 49.
FAQ
Why does a negative number have no real square root?
Any real number squared is positive or zero, so nothing real squares to a negative. The answer lies in imaginary numbers, written with i.
What is the difference between a square root and a cube root?
A square root is the value squared to reach n; a cube root is the value cubed to reach n. Cube roots accept negative inputs, square roots of negatives do not.
Is the square root of a perfect square always a whole number?
Yes. Perfect squares like 16, 81, and 144 have exact whole-number roots. All other positive integers have irrational roots.
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