Square Root Calculator
Enter any number to find its square root, cube root, or nth root — with exact answers and simplified radical form for perfect powers.
Whole number from 2 to 100
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Enter a number above to see its root
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Square roots, cube roots, and nth roots — how this calculator works
A square root asks "what number, multiplied by itself, gives this?" A cube root asks the same question three times over, and an nth root generalizes it to any number of multiplications. Doing this by hand is easy for small perfect powers (everyone remembers that 7 × 7 = 49) but gets tedious fast for anything else — and simplifying an "ugly" root like √72 into its cleanest form (6√2) takes a prime-factoring step most people don't bother with. This calculator does both: it gives the decimal answer to whatever precision you choose, and — for whole numbers — the exact simplified radical form, all computed instantly in your browser.
The math behind the results
- Square root (√x) — the value y such that y × y = x. Written with degree 2.
- Cube root (∛x) — the value y such that y × y × y = x. Unlike square roots, cube roots of negative numbers are real (∛-8 = -2).
- Nth root — the value y such that y raised to the power n equals x, for any whole-number degree from 2 to 100.
To simplify a radical, the calculator factors the number under the root into primes, then pulls out every group of "degree" matching primes as a whole number in front of the root sign. For √72: 72 = 2³ × 3², which regroups as (2 × 3)² × 2 = 6² × 2, so √72 = 6√2. The same factoring works for cube roots and any nth root — it just looks for groups of 3, 4, or however many primes the degree calls for instead of groups of 2.
Example 1 — simplifying a square root. √72: since 72 = 36 × 2 and 36 is a perfect square, √72 = 6√2 ≈ 8.4853. Enter 72 in square root mode to see both forms side by side.
Example 2 — simplifying a cube root. ∛24: since 24 = 8 × 3 and 8 is a perfect cube (2³), ∛24 = 2∛3 ≈ 2.8845.
Example 3 — an nth root. The 4th root of 48: since 48 = 16 × 3 and 16 is a perfect 4th power (2⁴), the 4th root of 48 = 2 × the 4th root of 3 ≈ 2.6321. Switch to nth root mode, set the degree to 4, and enter 48 to check it.
Perfect squares from 1 to 25
Recognizing perfect squares makes simplifying radicals much faster. Here is every perfect square up to 25² for quick reference:
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 10 | 100 | 19 | 361 |
| 2 | 4 | 11 | 121 | 20 | 400 |
| 3 | 9 | 12 | 144 | 21 | 441 |
| 4 | 16 | 13 | 169 | 22 | 484 |
| 5 | 25 | 14 | 196 | 23 | 529 |
| 6 | 36 | 15 | 225 | 24 | 576 |
| 7 | 49 | 16 | 256 | 25 | 625 |
| 8 | 64 | 17 | 289 | ||
| 9 | 81 | 18 | 324 |
Perfect cubes from 1 to 10
| n | n³ |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
Negative numbers and imaginary results
Square roots (and any even-degree root) of a negative number are not real numbers, because squaring — or raising to any even power — always produces a non-negative result, so it can never "undo" back into a negative. Mathematicians instead use imaginary numbers, defined so that i = √-1: the square root of -16 becomes 4i. Cube roots and other odd-degree roots don't have this problem — ∛-27 = -3 is a perfectly ordinary real number. This calculator shows the imaginary value automatically in square root mode, and a plain-language note for even-degree nth roots.
Common mistakes and quick tips
- "The" square root vs. two square roots: every positive number has a positive and a negative square root (both 4 and -4 square to 16), but "the square root" conventionally means the positive (principal) one, which is what this calculator returns.
- Simplified form only applies to whole numbers: radical simplification (like 6√2) is only meaningful for integer inputs — decimals show a rounded decimal answer only, since there's no clean radical form to factor out.
- Very large numbers: for performance and precision, this calculator accepts values up to ±1,000,000,000,000,000 and skips radical simplification above one trillion, showing a precise decimal result instead.
Sources & further reading
Frequently asked questions
What is the square root of a number?
The square root of a number x is the value that, when multiplied by itself, gives x. For example, the square root of 16 is 4, because 4 × 4 = 16. Every positive number technically has two square roots — a positive one and a negative one — but this calculator shows the principal (positive) root, which is the one used in everyday math and the one textbooks and search results mean by default.
How do you simplify a square root like √72?
To simplify a square root, split the number under the radical into a perfect-square factor and a leftover factor: 72 = 36 × 2, and 36 is a perfect square (6²), so √72 = √36 × √2 = 6√2. This calculator does that factoring automatically for any whole number — enter 72 in square root mode and it shows the exact simplified form 6√2 alongside the decimal approximation 8.4853.
What is the square root of a negative number?
No negative number has a real square root, because multiplying any real number by itself — positive by positive, or negative by negative — always gives a positive result; it can never equal a negative number. Mathematicians handle this with imaginary numbers: the square root of -16, for instance, is written as 4i, where i is defined as the square root of -1. This calculator detects negative input in square root mode and shows that imaginary value automatically.
How is a cube root different from a square root?
A cube root asks which number, multiplied by itself three times, gives the original number — and unlike square roots, cube roots of negative numbers are real. For example, the cube root of -8 is -2, because -2 × -2 × -2 = -8. Even-degree roots (square, 4th, 6th …) of negative numbers have no real answer, but odd-degree roots (cube, 5th, 7th …) always do, which is why this calculator only shows a "no real root" note for even root degrees.
Is the number I enter sent to a server?
No. This square root calculator runs entirely in your browser — the number, root type, and precision you choose never leave your device, and nothing is uploaded or stored on a server. There's no sign-up and no tracking, so you can use it as often as you like, including for homework or quick checks, with your input staying completely private.