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Square Root Calculator
Use this free square root calculator to find the square root of any number — with the simplified surd form when one exists (like √72 = 6√2), a clear note when the number is a perfect square, and a built-in check that squares the answer back. It accepts decimals as well as whole numbers, explains the method, and needs no sign-up.
How to use this calculator
- Type the number in the box — a whole number like 72, or a decimal like 0.25.
- Read the result: the square root to four decimal places, updated as you type.
- Check the line below the answer: it shows the simplified surd form (6√2) when the number simplifies, or tells you the number is a perfect square.
- For cube roots and other roots, use the nth root and cube root tools linked below.
Squares and square roots 1 to 25
| Number (n) | n² | √n | Number (n) | n² | √n |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 14 | 196 | 3.7417 |
| 2 | 4 | 1.4142 | 15 | 225 | 3.873 |
| 3 | 9 | 1.7321 | 16 | 256 | 4 |
| 4 | 16 | 2 | 17 | 289 | 4.1231 |
| 5 | 25 | 2.2361 | 18 | 324 | 4.2426 |
| 6 | 36 | 2.4495 | 19 | 361 | 4.3589 |
| 7 | 49 | 2.6458 | 20 | 400 | 4.4721 |
| 8 | 64 | 2.8284 | 21 | 441 | 4.5826 |
| 9 | 81 | 3 | 22 | 484 | 4.6904 |
| 10 | 100 | 3.1623 | 23 | 529 | 4.7958 |
| 11 | 121 | 3.3166 | 24 | 576 | 4.899 |
| 12 | 144 | 3.4641 | 25 | 625 | 5 |
How it works
The square root of a number is the value that gives the number when multiplied by itself. √9 is 3 because 3 × 3 = 9, and √72 is about 8.4853 because 8.4853 × 8.4853 ≈ 72. Geometrically, it is the side length of a square whose area is the number — that picture is often the fastest way to understand what a root means.
Some roots come out exact. Numbers that are whole numbers multiplied by themselves — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 and so on — are called perfect squares, and their roots are whole numbers. Every other positive number has a root that is an irrational decimal: it never ends and never repeats, so we quote it to a few decimal places, like √2 ≈ 1.4142.
Many roots can still be written neatly as surds. The trick is to factor out the largest perfect square: 72 = 36 × 2, and since √36 = 6 exactly, √72 = 6√2. School exams in India usually want this simplified form rather than a decimal, so this calculator shows it whenever it exists. In the same way, √50 = 5√2, √12 = 2√3 and √18 = 3√2.
Negative numbers have no real square root, because a number multiplied by itself is never negative — (−3) × (−3) = +9, not −9. Their roots exist only as imaginary numbers built on i = √−1, which is a later topic; enter a value of 0 or more here. Zero itself is easy: √0 = 0.
Worked example
A student is asked to simplify √72 and give its decimal value.
Step 1: Factor 72 and look for a perfect square — 72 = 36 × 2, and 36 is a perfect square.
Step 2: Take the root of the square part — √36 = 6.
Step 3: Bring it outside the root sign — √72 = 6√2.
Step 4: For the decimal, use √2 ≈ 1.41421 — 6 × 1.41421 = 8.4853 (rounded to four places).
Check the answer by squaring it: 8.4853² ≈ 72.00, so the root is right. That is exactly what this calculator shows for 72 — the decimal 8.4853, the surd 6√2, and the check — and for a perfect square like 144 it simply reports 12, exactly.
FAQ
What is the square root of 72?
The square root of 72 is about 8.4853. In simplified surd form it is 6√2, because 72 = 36 × 2 and √36 = 6, so √72 = 6 × 1.41421 = 8.4853. This calculator shows both forms.
How do you simplify a square root?
Factor the number and pull out perfect squares. Write 72 as 36 × 2; the square root of 36 is exactly 6, so √72 becomes 6√2. If nothing factors out — for example √7 or √13 — the surd is already in its simplest form.
What is a perfect square?
A perfect square is a whole number multiplied by itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and so on. Their square roots are exact whole numbers — √144 = 12 — while every other number has a square root that is a never-ending decimal.
Can you find the square root of a negative number?
Not as a real number. Any real number multiplied by itself is positive or zero, so no real number squares to a negative. Square roots of negatives belong to the imaginary numbers (√−1 = i), and this calculator will ask you for a number of 0 or more.
What is the square root of 0.25?
The square root of 0.25 is 0.5, because 0.5 × 0.5 = 0.25. Decimals work just like whole numbers here: this calculator accepts any value of 0 or more, including fractions like 0.25 and 2.5 (whose root is about 1.5811).
How can I estimate a square root without a calculator?
Bracket it between two perfect squares. √72 sits between √64 = 8 and √81 = 9, and since 72 is closer to 64 it is a bit under 8.5 — the exact value is 8.4853. Refine by squaring your guess: 8.5² = 72.25, a touch high, so the true root is slightly less than 8.5.