What is a square root?
The square root of a number n is a value that produces n when multiplied by itself. For example, the square root of 49 is 7 because 7 × 7 = 49.
The symbol √ is called the radical sign. The expression √n means the principal, or non-negative, square root of n.
Square root formula
√n = x when x² = n
Example: √144 = 12 because 12² = 144.
How to calculate a square root
- Enter a non-negative number in the calculator.
- Select Calculate square root.
- Read the decimal result.
- If the number is a perfect square, the result will be a whole number.
- If it is not a perfect square, the result will be a decimal.
Square root examples
√144 = 12
√25 = 5
√2 ≈ 1.4142
√12 = 2√3 ≈ 3.4641
√50 = 5√2 ≈ 7.0711
How to simplify a square root
To simplify a square root, find the largest perfect square that divides evenly into the number. Then take its square root outside the radical.
√12
12 = 4 × 3
√12 = √4 × √3
√12 = 2√3
Another example:
√72
72 = 36 × 2
√72 = √36 × √2
√72 = 6√2
Square root of a fraction
To find the square root of a fraction, take the square root of the numerator and denominator separately.
√(a/b) = √a ÷ √b
√(9/16) = √9 ÷ √16 = 3/4
√(25/36) = 5/6
Square root of a negative number
A negative number does not have a real square root because no real number multiplied by itself gives a negative result. For example, √-1 is not a real number.
In advanced math, the square root of a negative number is written using the imaginary unit i. For example, √-1 = i and √-4 = 2i. For complex-number calculations, use the Complex Number Calculator.
Common square roots
| Number | Square root | Type |
|---|---|---|
| 1 | 1 | Perfect square |
| 4 | 2 | Perfect square |
| 9 | 3 | Perfect square |
| 16 | 4 | Perfect square |
| 25 | 5 | Perfect square |
| 36 | 6 | Perfect square |
| 49 | 7 | Perfect square |
| 64 | 8 | Perfect square |
| 2 | 1.4142 | Irrational |
| 3 | 1.7321 | Irrational |
| 5 | 2.2361 | Irrational |
| 10 | 3.1623 | Irrational |
Perfect squares list (1 to 30)
A perfect square is the result of multiplying a whole number by itself. Here are the perfect squares from 1² to 30² with their square roots.
| Perfect square | Square root |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
| 121 | 11 |
| 144 | 12 |
| 169 | 13 |
| 196 | 14 |
| 225 | 15 |
| 256 | 16 |
| 289 | 17 |
| 324 | 18 |
| 361 | 19 |
| 400 | 20 |
| 441 | 21 |
| 484 | 22 |
| 529 | 23 |
| 576 | 24 |
| 625 | 25 |
| 676 | 26 |
| 729 | 27 |
| 784 | 28 |
| 841 | 29 |
| 900 | 30 |
How to find a square root without a calculator
Find the two perfect squares your number sits between, then estimate. For √50, 7² = 49 and 8² = 64, so √50 is just over 7. Since 50 is only 1 more than 49, the answer is about 7.07, and the exact value is 7.0711 (rounded). You can also write square roots with exponents: √n = n1/2.
Where square roots are used
Square roots are used in algebra, geometry, physics, statistics and construction. Common examples include finding the side length of a square from its area, calculating distances with the Pythagorean theorem, and working with quadratic equations.
Frequently asked questions
What is the square root of a perfect square?
A perfect square has a whole-number square root. For example, √100 = 10 and √81 = 9.
Can I calculate the square root of a decimal?
Yes. Enter decimals such as 2.25, 0.64 or 12.5. The calculator will return the decimal square root.
What is the square root of -1?
In real numbers, √-1 is undefined. In complex numbers, √-1 is represented by the imaginary unit i.
How do I simplify √12?
Find the largest perfect square factor: 12 = 4 × 3. Since √4 = 2, the simplified form is 2√3.
How do I find the square root of a fraction?
Take the square root of the numerator and denominator separately. For example, √(9/16) = 3/4.
Is every square root irrational?
No. Perfect squares have rational square roots. For example, √36 = 6. Numbers such as √2, √3 and √5 are irrational.
What is the difference between √ and ²?
√ means square root, while ² means squared. They are inverse operations: 7² = 49 and √49 = 7.
Can I use this calculator for homework?
Yes. It is useful for checking square roots, perfect squares, simplified radicals and decimal approximations.
How do I calculate a square root without a calculator?
Find the nearest perfect squares on either side of your number and estimate between them. For √50, 7² = 49 and 8² = 64, so the answer is just above 7 (about 7.07). For exact simplified answers, factor out the largest perfect square.