Enter a number above to calculate its square root.
Square Roots Explained
A square root is the number that, multiplied by itself, gives the original number: √144 = 12 because 12 × 12 = 144. Numbers that are not perfect squares have irrational roots, so √2 = 1.41421356 and the digits never end or repeat. Enter any number above to get its square, cube or nth root, the simplified radical and a check.
Every positive number has two square roots, one positive and one negative, because 12 × 12 and (−12) × (−12) both equal 144. The √ symbol always means the positive one, called the principal root. In exponent form √x = x1/2, the cube root is x1/3, and the nth root is x1/n, which is exactly how this calculator works out every root.
How to Find a Square Root by Hand
1. Bracket it between perfect squares
For √50, note that 7² = 49 and 8² = 64. So the answer is between 7 and 8, and much closer to 7.
2. Improve the guess with the Babylonian method
Take a guess g, divide the number by it, and average the two: new guess = (g + N ÷ g) ÷ 2. Each round roughly doubles the number of correct digits.
| Round | Calculation | New guess |
|---|---|---|
| Start | 7 (from step 1) | 7 |
| 1 | (7 + 50 ÷ 7) ÷ 2 | 7.0714286 |
| 2 | (7.0714286 + 50 ÷ 7.0714286) ÷ 2 | 7.0710678 |
Two rounds already give √50 = 7.0710678, correct to 7 decimal places.
3. Check by squaring
7.0710678 × 7.0710678 = 49.9999999, so the answer is right. Squaring your result is the quickest way to catch a slip.
Simplified Radicals and Decimal Values
To simplify, pull the largest perfect square factor out of the root: √72 = √(36 × 2) = 6√2. A number with no perfect square factor above 1, like 2, 3 or 10, is already in simplest form.
| Number | Simplified | Decimal | Number | Simplified | Decimal |
|---|---|---|---|---|---|
| 2 | √2 | 1.4142 | 24 | 2√6 | 4.8990 |
| 3 | √3 | 1.7321 | 27 | 3√3 | 5.1962 |
| 5 | √5 | 2.2361 | 32 | 4√2 | 5.6569 |
| 6 | √6 | 2.4495 | 45 | 3√5 | 6.7082 |
| 7 | √7 | 2.6458 | 48 | 4√3 | 6.9282 |
| 8 | 2√2 | 2.8284 | 50 | 5√2 | 7.0711 |
| 10 | √10 | 3.1623 | 72 | 6√2 | 8.4853 |
| 12 | 2√3 | 3.4641 | 75 | 5√3 | 8.6603 |
| 18 | 3√2 | 4.2426 | 98 | 7√2 | 9.8995 |
| 20 | 2√5 | 4.4721 | 200 | 10√2 | 14.1421 |
Negative Numbers, Decimals and Common Mistakes
Negative numbers
No real number squared gives a negative, so √(−9) is the imaginary number 3i, where i = √(−1). The calculator shows that form, and √(−72) as 6√2 i. Odd roots are different: the cube root of −8 is simply −2, because (−2)³ = −8. Even roots of 4 or higher of a negative number have no real value.
Zero, one and numbers between 0 and 1
√0 = 0 and √1 = 1. For a number between 0 and 1 the root is bigger than the number itself: √0.25 = 0.5 and √0.01 = 0.1. That is not an error, since multiplying two numbers below 1 always gives something smaller.
Mistakes to avoid
- Splitting a sum. √(9 + 16) = √25 = 5, not √9 + √16 = 7. Roots split over multiplication and division, never over addition or subtraction.
- Forgetting the negative root in equations. x² = 49 has two solutions, x = 7 and x = −7, even though √49 on its own is just 7.
- Squaring a negative in your head. √(x²) equals the absolute value of x, so √((−5)²) = 5, not −5.
Square roots turn up constantly in geometry. The diagonal of a square with side s is s√2, and the Pythagorean theorem calculator uses √(a² + b²) to find a missing side. For powers and fractional exponents, try the exponent calculator.
Perfect Squares Reference Table
Click any value to calculate its square root instantly.