# What is the square root of a number?

Finds the square root of a number with its simplified radical form, or the number from its square root.

- Page: https://www.acalculator.org/math/square-root-calculator
- JSON spec: https://www.acalculator.org/math/square-root-calculator.json
- Version: c57eefc45fd3

## Default answer

Example with the default inputs (Number (x) 72): The square root of 72 is 8.485281.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | Number (x) | The number to take the square root of, 0 or more. A whole number keeps every digit. |
| root | Or its square root (√x) | Leave the number empty and type a square root, 0 or more, to get the number back. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| root | Square root (√x) | The principal square root: the number that is 0 or more and gives x when squared. |
| number | Number (x) | The number, with every digit as typed (or worked out from the root). |
| negative | Negative square root | The other number whose square is x: −√x. |
| simplified | Simplified radical form | For a whole number, √x written as k√s with s as small as possible (√72 = 6√2). |
| perfect | Perfect square? | For a whole number, whether it is the square of a whole number. |

## Method

√x = r, where r ≥ 0 and r² = x.

## Assumptions

- Real numbers only: x must be 0 or more (the square root of a negative number is imaginary).
- The principal root: √x is the root that is 0 or more; −√x is the other square root.
- A whole number of up to 100 digits is read exactly: the perfect-square check and the root of a perfect square are exact at any such size.
- The simplified radical form is shown for whole numbers up to 10¹⁸, and for every perfect square.
- A decimal is read as a 64-bit float, about 16 significant digits.

## Worked examples

1. x = 72 gives root = 8.485281, negative = -8.485281, simplified = 6√2, perfect = no. Source: hand calculation in content.mdx: 72 = 36 × 2, so √72 = 6√2; Python 3: math.sqrt(72) = 8.48528137423857.
2. x = 144 gives root = 12, simplified = 12, perfect = yes. Source: hand calculation in content.mdx: 12² = 144.
3. x = 2 gives root = 1.414214, simplified = √2, perfect = no. Source: hand calculation in content.mdx: 2 has no square factor; Python 3: math.sqrt(2) = 1.4142135623730951.
4. root = 1.5 gives number = 2.25. Source: hand calculation in content.mdx: 1.5² = 2.25.
5. x = 0.25 gives root = 0.5. Source: hand calculation in content.mdx: 0.5² = 0.25.
6. x = 12345678987654321 gives root = 111,111,111, number = 12,345,678,987,654,321, simplified = 111111111, perfect = yes. Source: hand calculation in content.mdx: 111,111,111² = 12,345,678,987,654,321; Python 3: math.isqrt(12345678987654321) = 111111111.
7. x = 1800 gives root = 42.426407, simplified = 30√2. Source: hand calculation in content.mdx: 1,800 = 900 × 2, so √1,800 = 30√2; Python 3: math.sqrt(1800) = 42.42640687119285.

## FAQ

### What is a square root?

A square root of x is a number that gives x when you multiply it by itself. 5 × 5 = 25, so 5 is a square root of 25. So is −5, because (−5) × (−5) = 25 too. The symbol √ means the principal square root, the one that is 0 or more, so √25 = 5.

### How do I simplify a square root?

Find the largest perfect square that divides the number, and take its root outside the radical. 72 = 36 × 2 and √36 = 6, so √72 = 6√2. A square root is fully simplified when the number left inside has no square factor other than 1.

### What is a perfect square?

A whole number that is the square of a whole number: 1, 4, 9, 16, 25, 36 and so on. Its square root is a whole number, and its simplified radical form has nothing left inside the root.

### Can I take the square root of a negative number?

Not as a real number, because any real number times itself is 0 or more. The square roots of −9 are the imaginary numbers 3i and −3i, where i = √−1. This calculator works with real numbers, so it asks for a number of 0 or more.

### Why is the square root of 2 not a fraction?

Because √2 is irrational: no fraction of whole numbers squares to exactly 2. Its decimal goes on forever without repeating (1.41421356…). The same is true for the square root of every whole number that is not a perfect square.

### How do I find a square root without a calculator?

Find the two perfect squares either side of the number, then refine. For √30: 5² = 25 and 6² = 36, so the root is between 5 and 6. Try 5.5: 5.5² = 30.25, a little too big. Try 5.47: 5.47² = 29.92, a little too small. Each guess halves the gap, and √30 = 5.4772…

## Sources

- OpenStax, Elementary Algebra 2e, section 9.1 Simplify and Use Square Roots: https://openstax.org/books/elementary-algebra-2e/pages/9-1-simplify-and-use-square-roots
- OpenStax, Elementary Algebra 2e, section 9.2 Simplify Square Roots: https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots