What a square root is
The square root of a number is the value that, multiplied by itself, gives that number: √x. The square root of 144 is 12, because 12 × 12 = 144. Perfect squares (4, 9, 16, 25…) have whole-number roots; most numbers give a non-repeating decimal.
Square roots are the inverse of squaring, so they undo it: squaring 12 gives 144 and rooting 144 returns 12. Geometrically the root is the side length of a square whose area is your number, which is why a 144-unit square has 12-unit sides. For the third root rather than the second, use the cube root calculator.
How to use this calculator
Enter a non-negative number to get its square root. Negative numbers have no real square root (their roots are imaginary). A square root is the same as raising to the power ½ — see the exponent calculator.
The badge multiplies the answer by itself so you can confirm it lands back on your input. Results are rounded to six decimals, so non-perfect squares are approximations: √2 shows as 1.414214. Roots of numbers below 1 are larger than the number itself — √0.25 is 0.5.
Estimating roots by hand
You can bracket a root quickly using the perfect squares either side of it. For 150, note that 12 × 12 = 144 and 13 × 13 = 169, so the root must sit between 12 and 13, and much nearer 12 because 150 is close to 144.
Refine the guess by dividing and averaging: 150 ÷ 12 = 12.5, and the average of 12 and 12.5 is 12.25. Squaring that gives 150.06, already accurate to two decimals. Repeating the step converges fast, and it is the same method a calculator uses internally. The most common everyday use is finding the long side of a right triangle, which the hypotenuse calculator does in one step.
Frequently asked questions
- What is the square root of 144?
- 12, because 12 × 12 = 144.
- How do I calculate a square root?
- Find the number that multiplied by itself gives your value. For non-perfect squares, a calculator gives a decimal approximation.
- Can you take the square root of a negative number?
- Not in real numbers — negative numbers have imaginary square roots.
- Is a square root the same as a power of one half?
- Yes. √x equals x raised to the power 0.5.
- Why does an equation like x² = 144 have two answers?
- Because both 12 and −12 square to 144. The √ symbol means only the positive root by convention, so this tool returns 12; when solving an equation you must add the negative root yourself, writing x = ±12.