Square Root Calculator
Type a number to get its square root, rounded to the decimal places you choose. The calculator also shows the simplified radical form, whether the number is a perfect square, and the perfect squares on either side of it. Switch to cube root for negative numbers too.
Last updated November 3, 2026
Decimals are fine. A negative square root gives an imaginary answer.
From 0 to 10.
Result
The square root of 72 is about 8.4853.
| Measure | Value |
|---|---|
| Square root | 8.4853 |
| Simplified radical form | 6√2 |
| Perfect square? | No |
| Prime factors | 2^3 x 3^2 |
| Nearest perfect squares | 64 (8²) and 81 (9²) |
Roots use standard double precision, which is reliable to about 15 significant digits.
Show the work
Formula: √x is the number that gives x when multiplied by itself; for a negative x, √x = √|x| × i.
- You entered 72 and chose the square root.
- Bracket it: 8² = 64 and 9² = 81, so 72 lies between 64 and 81 and the root lies between 8 and 9.
- Decimal value: √72 = 8.485281374239, rounded to 4 decimal places = 8.4853.
- Prime factors of 72: 2^3 x 3^2.
- Each pair of equal primes comes out of the root: outside = 2 × 3 = 6; left inside = 2 = 2. So √72 = 6√2.
Result: √72 = 6√2 ≈ 8.4853
What a square root is and how it is found
√x = y means y × y = x
The square root of x is the number y that gives x when multiplied by itself. Because 12 × 12 = 144, the square root of 144 is 12. Every positive number has two square roots, one positive and one negative, since (-12) × (-12) is also 144. The √ symbol means the positive one, called the principal root.
Roots of whole numbers are either whole numbers or irrational. The square root of 2 is 1.4142135623730951 and its decimals never end or repeat, so any decimal answer is a rounded value. Roots of perfect squares such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and 169 are exact.
The calculator uses the square root built into your browser, which works in standard double precision and is reliable to about 15 significant digits.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Basic Calculator, Fraction Calculator and Exponent Calculator.
Simplifying a square root into radical form
A radical is simplified by pulling out every square factor. Write the number as a product of primes, then move each pair of equal primes outside the root. For 72 = 2³ × 3², the pair of 3s and one pair of 2s come out as 3 × 2 = 6, and a single 2 stays inside, so √72 = 6√2.
The same method gives √50 = 5√2, √12 = 2√3 and √200 = 10√2. If no prime appears twice, as in 15 = 3 × 5, the root is already in simplest form.
Radical form is shown for whole numbers up to 1,000,000,000,000. For decimals such as 0.25 the calculator gives the decimal root only.
Estimating a square root by hand
First bracket the number between two perfect squares. 72 sits between 64 (8²) and 81 (9²), so its root is between 8 and 9. It is 8 above 64 in a gap of 17, which suggests about 8 + 8 ÷ 17, or roughly 8.47. The exact value is 8.4853.
To sharpen a guess, average it with the number divided by the guess. Starting from 1 for the number 2, the average of 1 and 2 ÷ 1 is 1.5. Averaging 1.5 and 2 ÷ 1.5 gives 1.4167, and one more round gives 1.4142. This repeated averaging is a form of Newton's method and roughly doubles the number of correct digits each round.
Negative numbers and cube roots
No real number multiplied by itself is negative, so the square root of a negative number is imaginary. It is written with the imaginary unit i, defined by i × i = -1. That gives √-16 = 4i, because (4i) × (4i) = 16 × i² = -16. The calculator labels these answers as imaginary and shows the simplified form, for example √-72 = 6i√2.
A cube root asks for the number that gives x when multiplied by itself three times. It exists for every real number and keeps the sign, so the cube root of -27 is -3 because (-3) × (-3) × (-3) = -27. The cube root of 10 is about 2.1544, between 2 (since 2³ = 8) and 3 (since 3³ = 27).
How to use the Square Root Calculator
- Enter the number. Decimals are allowed, and a negative number gives an imaginary square root.
- Choose square root or cube root and the number of decimal places, from 0 to 10.
- Read the root, the simplified radical, the perfect-square check and the nearest perfect squares.
- Open Show the work to follow the steps, then Copy, Print or Share the result.
Worked example
Find the square root of 72 to 4 decimal places. The perfect squares around 72 are 64 (8²) and 81 (9²), so the root lies between 8 and 9. The decimal value is 8.485281…, which rounds to 8.4853. For the exact form, 72 = 2³ × 3². One pair of 2s and the pair of 3s come out, giving 2 × 3 = 6 outside, while one 2 stays inside. So √72 = 6√2. Check: 6 × 6 = 36, and 36 × 2 = 72.
| Number | Square root | Simplified | Perfect square |
|---|---|---|---|
| 2 | 1.4142 | √2 | No |
| 12 | 3.4641 | 2√3 | No |
| 50 | 7.0711 | 5√2 | No |
| 72 | 8.4853 | 6√2 | No |
| 144 | 12 | 12 | Yes |
| 0.25 | 0.5 | Decimal only | No |
Frequently asked questions
Is the square root of 2 a rational number?
No. The square root of 2 is irrational, so its decimal expansion never ends or repeats. Any decimal you see is a rounded value, and the exact answer is written as √2.
How do I know if a number is a perfect square?
Take its square root. If the result is a whole number, the number is a perfect square. This calculator checks whole numbers exactly and reports yes or no.
Why does a negative number give an answer ending in i?
Multiplying any real number by itself never gives a negative result, so mathematicians define i with i × i = -1. The square root of -9 is therefore 3i, an imaginary number.
How accurate are the decimals, and how large can the number be?
Results are reliable to about 15 significant digits. You can enter values up to 1,000,000,000,000,000 in size, and everything is calculated in your browser without being stored.