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How to Find a Square Root (and What It Really Means)
What a square root is, how to estimate one by hand, and the difference between square roots and cube roots — with worked examples and a calculator.
Updated 28 September 2026 · 3 min read
A square root answers a simple question: what number, multiplied by itself, gives you this? The square root of 144 is 12, because 12 × 12 = 144. That's the whole idea — a square root just undoes squaring.
The definition, in one line
Numbers like 1, 4, 9, 16, 25 and 144 are perfect squares — their roots are whole numbers. Most numbers aren't so tidy: √2 ≈ 1.414 and √10 ≈ 3.162 run on forever without repeating. That doesn't make them any less real; it just means you round to as many decimal places as you need.
Estimating a root without a calculator
You can get close to any square root by squeezing it between two perfect squares you already know:
- Find the nearest perfect squares on each side. For √50, that's 49 (7²) and 64 (8²), so the answer is a little more than 7.
- Since 50 is only just above 49, the root is only just above 7 — around 7.07.
- To sharpen it, divide the number by your guess and average the two: 50 ÷ 7.07 ≈ 7.07, so 7.07 is already spot on.
That last step is an old trick sometimes called the divide-and-average method, and it converges fast. For an exact figure with no arithmetic, the square root calculator gives the root to six decimal places instantly.
Can you take the root of a negative number?
Not within the ordinary (real) numbers. No real number times itself gives a negative result, because a negative times a negative is positive. So √(−9) has no real answer — such roots live among the 'imaginary' numbers, a topic for another day. Our calculator flags negative inputs rather than pretending they have a real root.
Square roots vs cube roots
A cube root (∛) undoes cubing instead of squaring: ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots of negatives are perfectly fine — ∛(−27) = −3, since (−3) × (−3) × (−3) = −27. The square root calculator shows both roots side by side, along with the number's square and cube, so you can see the whole family at once.
Where square roots actually show up
- Geometry. The length of a square's diagonal, or the hypotenuse of a right triangle via Pythagoras, both come from square roots.
- Statistics. Standard deviation is the square root of the variance, which is why 'root' appears throughout data work.
- Everyday scaling. If you double the area of a square, its side only grows by √2 (about 41%), not double.
Powers and roots are inverses, so it helps to keep the two together. Once you're comfortable with roots, the exponent calculator handles the squaring, cubing and higher powers that go the other way. For a refresher on the arithmetic that underpins all of this, our guide to percentages and the fraction calculator walkthrough are good next reads.
Frequently asked questions
- Does every number have two square roots?
- Positive numbers technically have two — one positive and one negative — because both 5 × 5 and (−5) × (−5) equal 25. By convention, 'the square root' (√) means the positive one, which is what calculators report.
- Why isn't the square root of 2 an exact decimal?
- Because √2 is irrational: it can't be written as a fraction of two whole numbers, and its decimal expansion never ends or repeats. You round it to as many places as your task needs — 1.41 is fine for most everyday work.