Square Root & Cube Root Calculator
Square Root Calculator (with Cube Root) — Instant, Accurate Results
Type in a number, and this square root calculator gives you the exact answer in a fraction of a second — no manual long division, no guessing. It also handles cube roots, so if you’re checking √52, ∛64, or anything in between, you’re covered. Below the calculator, you’ll find a plain-English explanation of what square and cube roots actually mean, two manual methods for working them out by hand, a quick-reference table for the numbers people search for most, and answers to the questions that come up again and again — including what happens when you try to take the square root of a negative number.
What Is a Square Root, Really?
The square root of a number n is the value that, multiplied by itself, gives you back n. Written with the radical symbol, √9 = 3, because 3 × 3 = 9. Every positive number actually has two square roots — a positive one and a negative one (both 3 and −3, squared, equal 9) — but when people ask “what’s the square root of 9?” they almost always mean the positive, or principal, root. That’s the value this calculator returns.
A cube root works the same way, just one step further: the cube root of n is the value that, multiplied by itself three times, gives you n. So ∛27 = 3, because 3 × 3 × 3 = 27. Unlike square roots, cube roots of negative numbers are perfectly normal in real numbers — ∛−27 = −3 — which is a detail we’ll come back to below.
How to Use the Calculator
Type your number into the input box, choose Square Root or Cube Root from the dropdown, and hit calculate. The result appears instantly, shown to six decimal places for numbers that don’t resolve to a whole number. There’s nothing to install and nothing to sign up for — it runs entirely in your browser.
Working It Out by Hand — Two Methods Worth Knowing
A calculator is faster, but understanding the method behind it makes the number make sense — and it’s genuinely useful when you don’t have your phone handy.
Method 1: Prime Factorization (for perfect squares)
If a number is a perfect square — meaning its square root is a whole number — you can find that root by breaking the number down into prime factors and pairing them up. Take 144: it factors into 2 × 2 × 2 × 2 × 3 × 3, which is (2×2) × (2×2) × (3×3). Pull one number out of each pair — 2, 2, 3 — and multiply them: 2 × 2 × 3 = 12. So √144 = 12. This works cleanly for any perfect square (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on) and is the fastest mental-math route once you’ve memorized the first dozen or so.
Method 2: The Estimation (Averaging) Method — for everything else
Most numbers aren’t perfect squares, so their roots are irrational — decimals that never terminate. To estimate √52 by hand: find the nearest perfect squares (49 and 64), so you know the answer sits between 7 and 8, closer to 7. Guess 7.2, divide 52 by that guess (52 ÷ 7.2 ≈ 7.22), then average your guess and the result: (7.2 + 7.22) ÷ 2 = 7.21. Repeat once more and you’ll land within a rounding error of the true value, 7.211103. This is essentially the same logic (Newton’s method) that calculators use internally — you’re just doing a couple of rounds of it by hand.
Squaring a number and finding its square root are opposite operations — it’s easy to reach for the wrong one under time pressure. If your answer to a square-root question comes out larger than the number you started with (for any number greater than 1), you’ve made an error somewhere; the square root of a number above 1 is always smaller than the number itself.
Square Roots of Popular Numbers — Quick Reference
Not every number is a tidy perfect square. Here are the exact or precise decimal values for the numbers people look up most often, so you can check your work or skip the calculation entirely.
| Expression | Result | Type |
|---|---|---|
| Square root of 2 | 1.414214 | Irrational |
| Square root of 5 | 2.236068 | Irrational |
| Square root of 8 | 2.828427 | Irrational |
| Square root of 48 | 6.928203 | Irrational |
| Square root of 52 | 7.211103 | Irrational |
| Square root of 72 | 8.485281 | Irrational |
| Square root of 75 | 8.660254 | Irrational |
| Square root of 81 | 9 | Perfect square |
| Square root of 289 | 17 | Perfect square |
Why 81 and 289 Are Different
√81 = 9 and √289 = 17 exactly, with no decimals, because 81 (9×9) and 289 (17×17) are both perfect squares. That makes them the easiest kind of square root problem — and also why they show up so often in textbooks and mental-math drills.
What About the Square Root of a Negative Number?
Try to take the square root of −4 and you’ll hit a wall in real numbers: no real number, multiplied by itself, produces a negative result (a negative times a negative is always positive). Mathematicians solve this with the imaginary unit, i, defined so that i² = −1. Under that system, √−4 = 2i. You’ll meet this in algebra, electrical engineering, and signal processing — but for everyday arithmetic, a negative number simply has no real square root, which is exactly what a well-built calculator should tell you rather than silently guessing.
Cube roots don’t have this problem. ∛−4 ≈ −1.587401, because a negative number multiplied by itself three times stays negative. If you’re working with volumes, densities, or anything that can genuinely be a negative quantity, the cube root will always give you a real, usable answer.
Where Square and Cube Roots Actually Get Used
These aren’t just classroom exercises. A few places roots show up in real work:
- Construction & carpentry: the Pythagorean theorem (a² + b² = c²) uses square roots to find a diagonal measurement — checking that a wall, deck, or foundation is truly square.
- Finance: standard deviation, a core measure of investment risk, is calculated as the square root of variance.
- Physics: formulas for free-fall time, pendulum period, and kinetic energy all involve square roots.
- Engineering & design: cube roots are used to find the side length of a cube from its volume — useful for packaging, tank sizing, and 3D printing calculations.
- Statistics & data science: root-mean-square (RMS) calculations, used to measure signal strength and error rates, rely on square roots at their core.
Why Use a Calculator Instead of Doing It by Hand?
For perfect squares under 400 or so, mental math and memorization get you there quickly. Past that — or for anything that isn’t a perfect square — manual estimation takes several rounds to get precise, and it’s easy to make a small arithmetic slip that throws off the whole answer. A dedicated calculator gives you the exact value (or six decimal places of precision for irrational roots) instantly, works the same way on a phone as a laptop, and validates your input so a typo doesn’t produce a nonsense result.
Our other free math tools cover the calculations that usually come up alongside square roots.
Frequently Asked Questions
Exactly 9. 81 is a perfect square (9 × 9 = 81), so its square root is a whole number with no decimals.
Exactly 17. Like 81, 289 is a perfect square (17 × 17 = 289).
There’s no real number answer — no real number multiplied by itself gives a negative result. In terms of imaginary numbers, √−4 = 2i, where i² = −1. For everyday, real-number arithmetic, treat it as undefined.
Approximately 7.211103. Since 52 isn’t a perfect square, this value is irrational and continues indefinitely without repeating.
Approximately 8.485281. You can simplify it as 6√2, since 72 = 36 × 2 and √36 = 6.
Approximately 8.660254. Simplified, it’s 5√3, since 75 = 25 × 3 and √25 = 5.
Approximately 6.928203. Simplified, it’s 4√3, since 48 = 16 × 3 and √16 = 4.
Yes — unlike square roots, cube roots of negative numbers are real. For example, ∛−8 = −2, because −2 × −2 × −2 = −8.
Results are shown to six decimal places for numbers that aren’t perfect squares or perfect cubes, which is precise enough for schoolwork, professional estimates, and most engineering checks.
No — it runs directly in your browser on desktop or mobile, with nothing to install and no account required.
Final Thoughts
Square and cube roots aren’t complicated once you see what they’re actually asking — “what number, multiplied by itself, gets me here?” Whether you’re checking a perfect square like 81 or 289, estimating an irrational one like √52 or √72, or running into the special case of a negative number, the calculator above gives you the precise answer immediately, and the methods on this page let you sanity-check it by hand whenever you want to.
Further reading: Math Is Fun — Square Roots and Wikipedia — Nth Root for a deeper look at root notation and imaginary numbers.