Square root calculator
Type a number, choose the kind of root and see the answer at once. The tool works out the square root (√), the cube root (∛) or an nth root with a degree of your choice. If the answer is not a whole number, you see the simplified form where one exists, and a check that raises the answer back to the power.
Decimals may use a point or a comma. A sum works too, such as 2^10 or sqrt(2).
A whole number from 2 to 1,000. With 3 you get the cube root.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
What is a root?
The root of a number is the number you multiply by itself to get it. √49 = 7, because 7 × 7 = 49. For a cube root you multiply three times: ∛27 = 3, because 3 × 3 × 3 = 27. In general, the nth root of x is the number whose nth power equals x.
Examples
| Problem | Answer |
|---|---|
| √144 | 12 |
| √2 | ≈ 1.414213562 |
| √72 | 6√2 ≈ 8.485281374 |
| ∛27 | 3 |
| ∛−8 | −2 |
| 5th root of 32 | 2 |
| √−9 | no real number (3i) |
Simplifying roots
A root of a whole number can be simplified if the number has a square as a divisor. Find the largest one, take its root out in front and leave the rest under the root sign: √50 = √(25 × 2) = 5√2. That is useful for exact answers, for instance with Pythagoras or the quadratic formula. A square with side 1 has a diagonal of √2: about 1.414.
Where roots turn up
- The hypotenuse of a right triangle is √(a² + b²). See also the right triangle calculator.
- The solutions of a quadratic equation contain a root of the discriminant; work them out with the quadratic equation solver.
- The side of a square with area A is √A. A 50 m² square garden is therefore about 7.07 m wide.
- With compound growth you find the yearly rate with an nth root: if something grows 50% in 3 years, the yearly growth is ∛1.5 − 1 ≈ 14.5%.
Is it a power rather than a root you need? Use the exponent calculator.
Frequently asked questions
How do you work out a square root without a calculator?
Find the two perfect squares the number sits between and estimate the rest. √50 lies between √49 = 7 and √64 = 8, and much closer to 7, so about 7.07. The tool gives the exact answer to ten significant digits: 7.071067812.
What does √72 = 6√2 mean?
It is the same value in simplified form. Because 72 = 36 × 2 and 36 is a square (6 × 6), you take the 6 out in front of the root. The result 6√2 is exact; 8.485281374 is a rounding. If the answer is a whole number there is no simplified form, as it would be redundant.
Can I take the square root of a negative number?
Not among the real numbers, because a square is never negative. With complex numbers you can: √−9 = 3i. The tool says there is no real answer and names the complex one. A cube root may be negative: ∛−8 = −2.
How are roots related to powers?
The nth root of x is the same as x to the power 1/n. So √x = x^(1/2) and ∛x = x^(1/3). A root is a reversed power, and the check in the tool raises the answer to the power n again.