Square Root Calculator
Calculate the square root of any number with step-by-step simplification. Supports decimals, fractions, and exact radica
Calculation Type
Input Values
Common Perfect Squares
Square Root
√16
Exact Result
Simplified Radical
Factored form
Squared
(4.00)²
Reciprocal
0.250000
Radical Simplification
4 × √1
How it works
A square root answers “what number, multiplied by itself, gives this?” It's the inverse of squaring. Every positive number has two square roots (one positive, one negative); the calculator reports the principal (positive) one.
Square root
√n = x such that x² = n
- n
- the number (radicand)
- x
- the square root
Worked example
- Find √144
- Which x gives x² = 144?
- 12 × 12 = 144
√144 = 12.
Good to know
- Perfect squares (1, 4, 9, 16, 25…) have whole-number roots; everything else is irrational and only approximated.
- You can't take the real square root of a negative number — that requires imaginary numbers.
- √(ab) = √a · √b, which helps simplify roots by pulling out perfect-square factors.
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Frequently Asked Questions
What is a square root?
The square root of x is the number that multiplied by itself gives x. The principal (positive) root is the usual answer: √49 = 7, even though (-7)² is also 49.
Can I take the square root of a negative number?
Not within the real numbers — no real number squared is negative. Mathematicians extend to imaginary numbers, where √-9 = 3i, used in engineering and physics.
How do I simplify a square root?
Factor out the largest perfect square: √48 = √(16 x 3) = 4√3. Exact radical form like 4√3 is often preferred in algebra over a rounded decimal.
How do I estimate a square root by hand?
Bracket the number between perfect squares, then refine: √40 is between 6 (36) and 7 (49), closer to 6.3. Averaging a guess g with x/g (Newton's method) converges very fast.
Why are most square roots irrational?
Unless a whole number is a perfect square, its square root is irrational — the decimal never repeats or ends. That's why √2 ≈ 1.41421356... can only be written exactly in radical form.