Square Root Calculator
Find square roots, cube roots, and any nth root (plus the simplified radical form such as 6√2).
Details
Square root of 72
6√2
≈ 8.485281 – √72, the principal root
Square root (√x)
8.485281
Cube root (∛x)
4.160168
The number squared
5,184
Is it a perfect square?
No
This finds the square root of a number, or any other root you ask for. Choose square, cube, fourth or type your own index, then enter the number.
Where the answer is not a whole number it returns the simplified radical form as well as the decimal, so √72 comes back as 6√2 rather than just 8.485.
What is a square root?
A square root asks: which number, multiplied by itself, gives this?
Since 12 × 12 = 144, the square root of 144 is 12. Squaring and square-rooting undo each other, the way multiplying and dividing do.
The name comes from actual squares. A square with an area of 144 has sides of 12, so finding a square root is literally working backwards from the area of a square to the length of its side.
The tick-shaped symbol is the radical sign, and the number under it is called the radicand. A small number tucked into the notch, as in ∛, asks for a different root.
What to enter
- Which root do you want?
- Square root (multiplied by itself twice), cube root (three times), or any other root you name.
- Number
- The number under the root sign. Square roots of negative numbers have no ordinary answer, so those are shown as imaginary results.
What this assumes
The principal (positive) root is shown as the main answer, which is the standard convention.
How to calculate a square root
For perfect squares you can often just recall it. For everything else, estimating between two you know gets you very close.
- x
- The number under the root sign
- y
- The answer: what you multiply by itself
Check for a perfect square. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. If your number is on that list, the answer is exact and worth memorising.
Otherwise, trap it between two. For √50: 49 is 7² and 64 is 8², so the answer sits between 7 and 8, much nearer 7.
Refine by guessing and checking. Try 7.1: 7.1 × 7.1 = 50.41, slightly too big. Try 7.07: 49.98, very close. That is √50 to two decimals.
See a worked example: working out √144 without a calculator
- Number
- 144
Ask what times itself makes 144.
10 × 10 = 100, too small. 13 × 13 = 169, too big. So it is between 10 and 13.
Try 12: 12 × 12 = 144. Exact.
Cube roots work the same way but with three: ∛27 = 3, because 3 × 3 × 3 = 27.
√144 = 12
Frequently asked questions
Because multiplying any ordinary number by itself gives a positive result. Positive times positive is positive, and negative times negative is also positive. Nothing lands on a negative.
Mathematicians handle it with the imaginary unit i, where i² = −1. So √−9 is written 3i. You will meet this in the quadratic formula when the discriminant comes out negative.
Strictly, yes. Both 12 and −12 give 144 when squared.
The √ symbol means the positive one by convention, so √144 = 12. Both matter when solving an equation like x² = 144, where the answer is x = ±12.
A square root asks what number multiplied by itself twice gives your number. A cube root asks what gives it after multiplying three times.
√27 is about 5.196, while ∛27 is exactly 3. Cube roots can handle negatives, because −3 × −3 × −3 = −27.
Because most numbers are not perfect squares. √2 = 1.41421356… and continues forever without repeating.
These are called irrational numbers. It is often more accurate to leave an answer as √2 than to round it, which is why exam answers are frequently written in that form.
Problems people actually run into
Assuming √(a + b) equals √a + √b
It does not, and this is one of the most common errors in algebra. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.
Square roots do split across multiplication and division, so √(9 × 16) really is √9 × √16 = 12. Adding and subtracting are where it breaks.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: August 29, 2026