Square Root Calculator
The square root of 144 is 12 because 12 multiplied by 12 equals 144. This square root calculator finds the principal real square root of any non-negative number. It also reports the root with the opposite sign and checks whether the input has a whole-number square root. Enter an integer or decimal at least zero to get a result rounded to no more than 6 decimal places. Square roots reverse the operation of squaring. They appear in geometry, algebra, statistics, physics, and measurement problems. Perfect squares such as 0, 1, 4, 9, and 144 have exact whole-number roots. Other values usually produce decimals. The calculator uses real numbers only, so a negative input is outside its supported range.
Quick answer
The square root of a number is the value that, when multiplied by itself, gives that number.
What this tells you
- •The square root of a number is the value that, when multiplied by itself, gives that number.
- •Every positive number has two square roots, one positive and one negative, with the same size.
- •The principal square root is the non-negative one, so the square root of 144 is 12.
- •The negative root has the same magnitude with the opposite sign, so both 12 and -12 solve y squared equals 144.
- •Zero has one distinct real root, 0, because the positive and negative forms coincide.
- •The perfect-square check says Yes only when the computed root is an integer.
- •Results are rounded to no more than 6 decimal places, and unnecessary trailing zeros are not shown.
How to Use
- 1Enter the number you want the square root of in the Number field.
- 2Use a value of zero or greater, since negative numbers have no real square root.
- 3Click Calculate to see the principal square root, the root with the opposite sign, and the perfect-square check.
- 4Read the principal result as the standard meaning of the radical symbol. For example, the displayed square root of 81 is 9, not both 9 and -9.
- 5Use the negative-root line when solving an equation such as y squared equals 81, where both 9 and -9 satisfy the equation.
- 6If the result is decimal, remember that its displayed digits are rounded. Keep more precision in a longer calculation when the final answer depends on it.
How It Works
Formula
sqrt(x) = y, where y * y = xThe square root of x is a value y whose square equals x. The radical notation sqrt(x) means the principal, non-negative root. For x = 144, y = 12 because 12 * 12 = 144. When the question is instead to solve y * y = 144, both 12 and -12 work because (-12) * (-12) also equals 144. The calculator computes the root, rounds that value to 6 decimal places, and produces the negative result by changing its sign. It labels an input a perfect square when the unrounded square root is an integer. For sqrt(2), the full decimal continues without ending or repeating. Its displayed value is 1.414214, and squaring that rounded display gives a value close to 2 rather than exactly 2.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Square root of 144
Find the value that multiplies by itself to give 144. Since 12 * 12 = 144, the square root of 144 is 12. The number is a perfect square because 12 is a whole number.
Square root of 2
There is no whole number that multiplies by itself to give 2, so the result is irrational. Rounded to 6 decimal places, the square root of 2 is about 1.414214, which means 2 is not a perfect square.
Square root of 0
Zero multiplied by itself is zero, so sqrt(0) = 0. The opposite-sign result also displays as 0 because both signs represent the same root here. Zero is a perfect square since its square root is the integer 0.
Square root of 20.25
The principal square root is 4.5 because 4.5 * 4.5 = 20.25. The opposite root is -4.5. The calculator marks this input No for perfect square because its root is not an integer, even though the decimal square root is exact.
Square root of 1,000
The value lies between 31 and 32 because 31 squared is 961 and 32 squared is 1024. The calculator rounds sqrt(1000) to 31.622777. Its negative root is -31.622777, and 1000 is not a perfect square.
Perfect Squares and Their Roots
Common perfect squares and the whole number square root of each.
| Number | Square Root |
|---|---|
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 64 | 8 |
| 100 | 10 |
| 144 | 12 |
| 225 | 15 |
| 400 | 20 |
Zero is also a perfect square because 0 * 0 = 0. A value can have an exact decimal root, such as sqrt(20.25) = 4.5, without meeting this calculator's integer-root perfect-square check.
Principal roots, equation roots, and estimates
The radical symbol has a specific convention. Written on its own, sqrt(49) means 7, the principal non-negative root. It does not mean plus or minus 7. The plus-or-minus notation appears when solving an equation. If x squared equals 49, then x = 7 and x = -7 both satisfy the equation. Keeping these two contexts separate prevents a common sign error.
Perfect squares provide useful checkpoints for estimating an unfamiliar root. To estimate sqrt(70), locate 70 between 64 and 81. Their roots are 8 and 9, so sqrt(70) must be between 8 and 9. Since 70 is much closer to 64 than 81, a result near 8.37 is reasonable. This bounding check can catch a mistyped input or misplaced decimal point.
Decimal radicands follow the same rule. For example, sqrt(0.81) is 0.9 because 0.9 squared is 0.81. Moving a decimal point requires care because squaring changes place value by two powers of ten. The square root of 100 is 10, while the square root of 1 is 1 and the square root of 0.01 is 0.1.
Rounding matters when a root feeds another formula. The tool displays up to 6 decimal places. If you copy 1.414214 for sqrt(2) and square it, the product is only approximately 2 because the displayed number is rounded. Exact radical notation or higher internal precision may be better for symbolic work and sensitive numerical calculations.
Common mistakes
- Trying to take the square root of a negative number. Negative numbers have no real square root, so this calculator returns no result for values below zero.
- Confusing squaring with square rooting. Squaring multiplies a number by itself, while finding a square root reverses that step, so the two operations are not the same.
- Ignoring that every positive number has two roots. Both a positive and a negative value square to the same number, so 25 has the roots 5 and -5.
- Writing plus or minus for the principal radical. The expression sqrt(25) equals 5, while the solutions to x squared equals 25 are 5 and -5.
- Assuming every exact decimal root makes the input a perfect square. This tool reserves Yes for integer roots, so 20.25 has the exact root 4.5 but receives No.
- Rounding a root too early in a multistep problem. A six-decimal display is accurate for many checks, but repeated calculations can accumulate rounding differences.
- Estimating without checking nearby perfect squares. A claimed sqrt(70) outside the interval from 8 to 9 cannot be correct because 64 is less than 70 and 81 is greater.
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