Square Root Calculator
Enter any number to get its square root as a decimal and in simplest radical form, with the steps shown. Works with decimals, negative numbers (imaginary answers) and other roots such as the cube root. To raise a number to a power, use the exponent calculator.
Example loaded: 72. Change the number and select Calculate.
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Square root: the quick answer
The square root of a number is the value that gives the number when you multiply it by itself. The square root of 49 is 7, because 7 × 7 = 49. The square root of 2 is about 1.414214.
Every positive number has two square roots, one positive and one negative. The √ sign means the positive one, called the principal root.
Numbers such as 4, 9, 16 and 25 are perfect squares, so their roots are whole numbers. Most numbers are not, and their roots are decimals that never end, such as √2 and √3.
How to find a square root
- Check for a perfect square. If you know 12 × 12 = 144, then √144 = 12. The table below lists perfect squares up to 20² = 400.
- Simplify with prime factors. Break the number into primes and pair them up. 72 = 2 × 2 × 2 × 3 × 3 has a pair of 2s and a pair of 3s, so √72 = 2 × 3 × √2 = 6√2.
- Estimate by hand. For √30, 5² = 25 and 6² = 36, so the answer is between 5 and 6. Try 5.5, divide 30 by 5.5 to get 5.4545, then average the two: (5.5 + 5.4545) ÷ 2 = 5.477. Repeat to get closer. The true value is 5.477226.
- Use the calculator for any number, including decimals. It shows the decimal answer, the simplified radical and the steps.
For other roots, change the root box. A cube root (∛) asks which number multiplied by itself three times gives your number: ∛27 = 3.
Worked examples
- √144: 12 × 12 = 144, so the answer is 12.
- √50: 50 = 2 × 5 × 5, so √50 = 5√2 ≈ 7.071068.
- √0.25: 0.5 × 0.5 = 0.25, so the answer is 0.5.
- √−9: a negative number has no real root, so the answer is imaginary. √−9 = 3i, because (3i)² = −9.
- ∛−27: odd roots of negative numbers are real, so ∛−27 = −3, because (−3)³ = −27.
Perfect squares and common square roots
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Square roots that are not whole numbers, rounded to 4 decimals: √2 = 1.4142, √3 = 1.7321, √5 = 2.2361, √6 = 2.4495, √7 = 2.6458, √8 = 2.8284, √10 = 3.1623, √11 = 3.3166.
Frequently asked questions
How do I find the square root of a number?
Use the calculator above, or find a number that gives yours when multiplied by itself. For a perfect square such as 64 the answer is a whole number, 8. For other numbers, estimate between two perfect squares and refine, or simplify with prime factors.
What is the square root of 2?
About 1.41421356. It never ends and never repeats, so it is shown rounded. In radical form it stays as √2.
Can you take the square root of a negative number?
Not in the real numbers, because no real number multiplied by itself is negative. In complex numbers, √−9 = 3i, where i is defined by i² = −1. This calculator shows the imaginary answer.
What is a perfect square?
A whole number that is the square of another whole number: 1, 4, 9, 16, 25, 36, 49 and so on. Their square roots are whole numbers.
How do I simplify a square root?
Factor the number into primes, pair equal primes, and move each pair outside the root as one number. √72 = √(2² × 3² × 2) = 6√2. If no prime appears twice, the root is already in simplest form, like √10.
How do I find a square root without a calculator?
Find the two perfect squares around your number and guess a value between them. Then divide the number by your guess and average the two. Two or three rounds usually give 5 or 6 correct digits.
Why are there two square roots?
Both 8 × 8 and (−8) × (−8) equal 64, so 64 has two square roots, 8 and −8. The √ sign means the positive one only. The calculator shows both.
Is √(x²) always equal to x?
No. √(x²) is the positive value of x, written |x|. For x = −5, √25 = 5, not −5.
How do I find a cube root or another root?
Change the root box. 3 gives the cube root, 4 the fourth root, and so on up to 20. For example ∛64 = 4, because 4 × 4 × 4 = 64.
How accurate are the decimals?
Results are accurate to about 15 digits, which is the limit of the numbers a browser uses. You can show 2 to 12 decimal places.
About this calculator
Written by Umar Khan. The calculator uses the standard square root and nth root functions of your browser and checks whether the result is exact. Simplified radicals come from prime factorization of whole numbers up to 1,000,000,000,000,000. Your numbers stay in your browser. Last updated November 10, 2026. Questions or corrections: [email protected].
Written by Umar Khan
Umar Khan is a PhD student in engineering at NTNU, the Norwegian University of Science and Technology, in Norway. Every formula and worked example on this page is shown in full so you can check each result yourself. Corrections are welcome at [email protected].
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For information only, not professional advice.
Written by Umar Khan. Last updated: November 10, 2026.