Square Root Calculator
Type any number and get its square root instantly — to 8 decimals — with a built-in perfect-square checker.
Find a square root
Enter any non-negative number.
The cube root and nth root of your number appear under the square root — odd roots of negative numbers included.
- At the defaults, √144 = 12 exactly — a perfect square, so the check line reads “12 × 12 = 144” and 144 lights up in the perfect-squares grid below.
- The same number’s cube root and nth root (you pick the degree) are worked out alongside the square root.
- Not a perfect square? The root is shown rounded to 8 decimals and flagged as irrational or non-terminating.
Type any number and get its square root instantly — to 8 decimals — with a built-in perfect-square checker. Plus a complete 1²–25² reference table for the roots worth memorizing.
Perfect squares, 1² to 25²
The reference table — your input lights up if it is a perfect square.
Memorizing these 25 squares means you can spot perfect squares — and estimate roots — without any tool at all.
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Calculators for numbers, fractions and functions.
Find a square root in three steps
Type, read, understand.
Any non-negative value works.
Instant, to 8 decimals.
See if it is a perfect square.
⚙️The math behind square roots
The square root of x is the number y such that y² = x. Every positive number has two square roots (one positive, one negative); calculators return the principal root — the non-negative one.
Newton’s method: yₙ₊₁ = (yₙ + x ÷ yₙ) ÷ 2
Worked examples: √144 = 12 because 12 × 12 = 144 — a perfect square. √2 ≈ 1.41421356 — irrational, so its decimals never terminate or repeat; 1.41421356² = 1.99999999…
Because the square of any real number is never negative: a positive times a positive is positive, and a negative times a negative is positive too. So there is no real number y with y² = −4. Mathematicians extended the system with the imaginary unit i (where i² = −1), giving complex roots like √−4 = 2i. This tool stays in the real numbers and shows a friendly message instead.
A square root undoes squaring (y² = x); a cube root undoes cubing (y³ = x). ∛8 = 2 because 2³ = 8 — and unlike square roots, cube roots of negative numbers are real (∛−8 = −2). This page is a dedicated, distraction-free √ tool; for cube roots and nth roots, use the scientific calculator.
Frequently asked questions
What is the square root of 144?
12 — because 12 × 12 = 144. It is a perfect square, so the root is an exact whole number.
What is the square root of 2?
Approximately 1.41421356. √2 is irrational — its decimal expansion never ends or repeats — so every value you see is a rounded approximation.
How do you find a square root without a calculator?
Bracket it between perfect squares, then refine. For √50: 7² = 49 and 8² = 64, so the answer is just above 7 — about 7.07. For precision, use Newton’s method: start with a guess g, replace it with (g + x ÷ g) ÷ 2, and repeat. Two or three rounds give 8-decimal accuracy.
What is a perfect square?
A number made by squaring a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576 and 625 (that is 1² through 25²). Their square roots are exact integers — no decimals needed.
What is the cube root of 27, and how do nth roots work?
The cube root of 27 is 3 — because 3 × 3 × 3 = 27. An nth root asks the same question with n factors: the 5th root of 32 is 2 because 2⁵ = 32. Use the degree box in the calculator to find the cube root and any nth root of the number you entered. Odd-degree roots of negative numbers are real (the cube root of −27 is −3); even-degree roots of negatives are not.
Can you take the square root of a negative number?
Not in the real numbers — no real y satisfies y² = −x for positive x. In complex numbers the answer uses i (√−9 = 3i). This calculator works with real numbers only and will tell you when an input has no real root.
What are the square roots of 0 and 1?
√0 = 0 and √1 = 1 — the two numbers that are their own square roots. They are the fixed points of squaring: the only values where y² = y.
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