How to use the Square Root Calculator
- Enter a number that is zero or greater.
- Choose the type of root: square, cube, fourth or fifth.
- Read the root as a decimal, the simplified radical form and whether the number is a perfect power.
- Use the radical form for exact answers in maths class.
Formula used
The nth root of x is the number that, raised to the power n, gives x. Simplified form: pull out the largest perfect-power factor, for example √72 = √(36 × 2) = 6√2.
Worked example
Here is the Square Root Calculator worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Number | 72 |
| Root | Square root (√) |
| Result | |
| Root | 8.4852814 |
| Simplified radical form | 6√2 |
| Perfect power? | No |
Finding a square root
The square root of 72 is 8.485281, because 8.485281 × 8.485281 is about 72. 72 is not a perfect square, so the decimal never ends. Perfect squares such as 144 have whole-number roots: √144 = 12.
Perfect squares come from squaring whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. The calculator tells you when your number is one of them.
Simplified radical form
To simplify a square root, factor out the largest perfect square: 72 = 36 × 2, so √72 = √36 × √2 = 6√2. The calculator does this automatically, giving 6√2. Another example: √50 = 5√2, and √45 = 3√5. A number with no square factor, such as 2, stays as √2.
For cube roots, look for perfect cubes: 54 = 27 × 2, so ∛54 = 3∛2. Mathematicians prefer radical form for exact answers, because it avoids rounding.
Other roots
The cube root of 27 is 3, and yes: 3³ = 27. The fourth root of 81 is 3, and the fifth root of 32 is 2. Cube roots of negative numbers do exist in real numbers (∛−8 = −2), but this calculator accepts only zero or positive numbers.
To find any root, you can also raise the number to a fractional power: the nth root of x is x^(1/n). The exponent calculator on this site does this.
Where roots are used
Square roots appear in geometry (the side of a square of a given area, the Pythagorean theorem), statistics (standard deviation), physics (speed, energy) and finance (volatility). To find the side of a square plot of 200 square metres, take √200 = 14.14 m.
For quick mental estimates, find the two perfect squares around your number: √72 lies between √64 = 8 and √81 = 9, and is close to 8.5.
Estimating square roots by hand
You can approximate a square root without a calculator. First find the perfect squares on either side: √72 lies between √64 = 8 and √81 = 9. Then refine with the averaging method: take a guess g, and compute (g + 72 ÷ g) ÷ 2. With g = 8.5, that is (8.5 + 8.4706) ÷ 2 = 8.4853, already close to the true value of 8.48528.
Repeating the step improves the answer quickly. This is the method behind the way computers calculate square roots. It also shows that a square root of a non-perfect square is an irrational number, which means its decimal never ends or repeats.
Frequently asked questions
Can you take the square root of a negative number?
Not in real numbers. The result is imaginary. This calculator accepts zero or positive numbers.
What is a perfect square?
A whole number whose square root is also a whole number, such as 1, 4, 9, 16, 25 and 36.
What is the radical form?
An exact way to write a root, such as 6√2, without rounding the decimal.
How do I find the cube root?
Choose the cube root option. The cube root of x is the number that, multiplied by itself three times, gives x.
Why is the radical form not shown for decimals?
Simplification works on whole numbers. For decimals, use the decimal result.
What is the square root of 2?
About 1.41421356. It is irrational, so its decimal never ends or repeats.
How can I estimate a square root without a calculator?
Find the nearest perfect squares, guess a value between them, then average your guess with the number divided by the guess. Repeat to improve it.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.