★ MATH · ROOTS · SQUARE ROOTS
√144 = 12 — 12 × 12 is 144, so 144 is a perfect square. Put in any number 0 or more and the calculator shows the square root to six decimal places, the simplified radical form, whether it is a perfect square, and the check — plus a reverse box that squares any number. This one is the SQUARE ROOT tool — one number in, its root out — the scientific-calculator page is the GENERAL SCIENTIFIC tool — trig, logs, powers and roots together — not a root specialist; the squares-cubes-roots-chart page is the LOOKUP CHART — squares, cubes and roots for 1 to 30 — not a calculator for any number; and the fraction-calculator page is the FRACTION tool — adding and multiplying fractions — not roots. No square-root calculator exists on this site — this one is the first.
A square root is the number that, multiplied by itself, gives the number you started with. On the defaults it is exact: √144 is 12, because 12 × 12 is 144. For numbers that are not perfect squares the answer is a simplified radical — √50 is 5√2 — and a decimal stopped at six places. Decimals are rounded to 6 places; simplified radicals are exact, and decimals are approximations for irrational roots.
It means 12 × 12 is 144, so 12 is the square root of 144 — the number that, multiplied by itself, gives 144. Because 144 is a perfect square, the answer is exact and whole: there is no decimal tail and no radical left over. The simplified form is just 12, and the check is the multiplication itself: 12 × 12 is 144, so the answer is right. Not every number is so tidy: √2 is 1.414214 to six decimals, and the simplified form stays √2, because 2 has no square factor to pull out.
The simplified radical is exact; the decimal is an approximation. For an irrational root like √2, the decimal never terminates — 1.414214 is the value stopped at six places, not the whole value. The simplified radical, √2 or 5√2, is the exact answer, and it is the one to carry into further working. For a perfect square like 144, the two agree exactly: 12 is both the simplified form and the decimal.
Simplifying means pulling square factors out. √50 is 5√2 because 50 is 25 × 2 and 25 is a perfect square: the 5 comes out of the radical and the 2 stays in. √72 is 6√2 the same way, because 72 is 36 × 2. If the number under the radical has no square factor — 2, 3, 7 — there is nothing to pull out, and the simplified form is the radical itself.
Check by squaring. Whatever the calculator says, multiply the answer by itself: the product must be the number you started with. For a perfect square the check is exact — 12 × 12 is 144. For an irrational root the check is approximate at six decimals — (1.414214)² is 2.000000 to the places shown — because the decimal was rounded. If the square of the answer is nowhere near the input, the input was mistyped.
On the defaults: the square root of 144.
And the other way: 12² is 144, and 1.5² is 2.25 — the reverse box squares any number, so you can check a root by squaring it.
The defaults are the number 144, a perfect square chosen so the answer is exact and whole. They are a starting point, not a lesson: your number may be anything 0 or more, and the calculator shows the simplified radical, the six-decimal value, whether it is a perfect square, and the check. Negative numbers have no real square root, so the calculator refuses them rather than inventing an answer. Put your number in, read the answer, and check the working before you use it.
These notes repeat in every explainer section because this page is built to be read start to finish: the simplified radical is exact and the decimal is an approximation; simplifying pulls square factors out; and the check is squaring the answer.
| Output | Formula | What it means |
|---|---|---|
| Number | input | 144. |
| Square root (simplified) | √x, simplified radical | 12. |
| Square root (6dp unrounded) | decimal, 6 places | 12.000000. |
| Perfect square? | exact when the radical simplifies away | Yes — a perfect square. |
| Check | answer × answer | 12 × 12 = 144. |
| Square a number | r × r | 12² = 144 (144.000000 unrounded). |
| Verdict | in words | √144 = 12 — 12 × 12 is 144, so 144 is a perfect square. |
| Reverse: Smallest whole root whose square is 144 or more | step | 12: 12² = 144 (144.000000 unrounded). |
| Reverse: Smallest whole root whose square is 200 or more | step | 15: 15² = 225 (225.000000 unrounded). |
| Reverse: Largest whole root whose square is 200 or less | step | 14: 14² = 196 (196.000000 unrounded). |
| Reverse: Smallest whole root whose square is 1000 or more | step | 32: 32² = 1024 (1024.000000 unrounded). |
| Reverse: Largest whole root whose square is 1000 or less | step | 31: 31² = 961 (961.000000 unrounded). |
| Reverse: Smallest whole root whose square is 50 or more | step | 8: 8² = 64 (64.000000 unrounded). |
| √144 | same model | √144 = 12 (12.000000 unrounded). |
| √2 | same model | √2 = √2 ≈ 1.414214. |
| √50 | same model | √50 = 5√2 ≈ 7.071068. |
| √72 | same model | √72 = 6√2 ≈ 8.485281. |
| √0 | same model | √0 = 0 (0.000000 unrounded). |
| √0.25 | same model | √0.25 = 0.5 (0.500000 unrounded). |
| √3 | same model | √3 = √3 ≈ 1.732051. |
| √10000 | same model | √10000 = 100 (100.000000 unrounded). |
| √7 | same model | √7 = √7 ≈ 2.645751. |
| √200 | same model | √200 = 10√2 ≈ 14.142136. |
| Negative input −4 | same model | Square root of a negative number.. |
| Non-numeric input | same model | Enter a number.. |
| Empty input | same model | Enter a number.. |
| Very large input 1e12 | same model | √1000000000000 = 1000000 (1000000.000000 unrounded); check: 1000000 × 1000000 = 1000000000000. |
| √1 | same model | √1 = 1 (1.000000 unrounded); check: 1 × 1 = 1. |
| √0.5 | same model | √0.5 = √2/2 ≈ 0.707107; check: (0.707107)² ≈ 0.5. |
| √2.25 | same model | √2.25 = 1.5 (1.500000 unrounded); check: 1.5 × 1.5 = 2.25. |
| √0.0001 | same model | √0.0001 = 0.01 (0.010000 unrounded); check: 0.01 × 0.01 = 0.0001. |
| √999983 (prime near a million) | same model | √999983 = √999983 ≈ 999.991500; check: (999.991500)² ≈ 999983. |
| √10 | same model | √10 = √10 ≈ 3.162278; check: (3.162278)² ≈ 10. |
| √144.5 | same model | √144.5 = 17√2/2 ≈ 12.020815; check: (12.020815)² ≈ 144.5. |
The arithmetic is exact on the numbers you type: simplified radicals are exact, and perfect squares come out whole — 12 × 12 is 144. Decimals are rounded to 6 places, so for irrational roots the decimal is an approximation: 1.414214 squared is 2.000000 to the places shown, not exactly 2. Very large inputs are fine up to a point — 1,000,000,000,000 gives 1,000,000 — but beyond that the display rounds, and the simplified radical is the exact answer to carry forward. Negative numbers have no real square root, so the calculator refuses them. Treat the radical as the answer and the decimal as its working form.
√144 = 12 — 12 × 12 is 144, so 144 is a perfect square. √2 = 1.414214; √50 = 5√2 ≈ 7.071068; √72 = 6√2 ≈ 8.485281; √0.25 = 0.5; √3 = 1.732051; √7 = 2.645751. Smallest whole root whose square is 144 or more: 12. Smallest whole root whose square is 200 or more: 15. Largest whole root whose square is 200 or less: 14. Smallest whole root whose square is 1000 or more: 32. Largest whole root whose square is 1000 or less: 31. Smallest whole root whose square is 50 or more: 8. Source: Bright Side Kit, brightoolbox.com.
The square root of 144 is 12, because 12 × 12 is 144. 144 is a perfect square, so the answer is exact and whole — the simplified form is just 12, and the six-decimal value is 12.000000.
√50 simplifies to 5√2. 50 is 25 × 2, and 25 is a perfect square, so the 5 comes out of the radical and the 2 stays in. The decimal value is 7.071068 to six places.
Because its decimal never terminates or repeats — 1.414214 is the value stopped at six places, not the whole value. The exact answer is the radical √2 itself. No fraction of whole numbers equals √2 exactly.
Square the answer: the product must be the number you started with. 12 × 12 is 144, so √144 is 12. For an irrational root the check is approximate at six decimals — (1.414214)² is 2.000000 to the places shown.
The square root of 0.25 is 0.5, because 0.5 × 0.5 is 0.25. A quarter is a perfect square, so the answer is exact.
Not as a real number — the calculator refuses negative inputs. No real number multiplied by itself gives a negative result. Negative roots exist only as complex numbers, which this calculator does not do.
1,000,000 — a trillion is a perfect square, 1,000,000 × 1,000,000. The calculator handles very large inputs, and the simplified radical is the exact answer to carry forward.
72 is 36 × 2, and 36 is a perfect square, so the 6 comes out of the radical. √72 is 6√2, which is 8.485281 to six decimal places.