How to Use the Square Root Calculator
This tool allows you to find the root of any positive or negative number quickly.
Simply select the type of root you need (Square, Cube, or Nth) and enter the number in the input field.
The result will appear instantly on the right.
If you select Square Root, the tool calculates the value that, when multiplied by itself, equals your input ($x \cdot x = n$).
If you select Cube Root, it finds the value that, when multiplied by itself twice, equals your input ($x \cdot x \cdot x = n$).
For complex roots, use the Nth Root tab to specify the degree (e.g., 4th root, 5th root, etc.).
Understanding Square Roots and Perfect Squares
A square root of a number $x$ is a number $y$ such that $y^2 = x$; in other words, a number $y$ whose square is $x$.
For example, 4 and −4 are square roots of 16 because $4^2 = (-4)^2 = 16$.
Every non-negative real number $x$ has a unique non-negative square root, called the principal square root, which is denoted by $\sqrt{x}$,
where the symbol $\sqrt{}$ is called the radical sign or radix.
A number that is the square of an integer is called a perfect square. Below is a list of common perfect squares.
| Number (n) |
Square ($n^2$) |
Square Root ($\sqrt{n}$) |
| 1 | 1 | 1 |
| 2 | 4 | 1.414 |
| 3 | 9 | 1.732 |
| 4 | 16 | 2 |
| 5 | 25 | 2.236 |
| 6 | 36 | 2.449 |
| 7 | 49 | 2.645 |
| 8 | 64 | 2.828 |
| 9 | 81 | 3 |
| 10 | 100 | 3.162 |
| 11 | 121 | 3.316 |
| 12 | 144 | 3.464 |
| 13 | 169 | 3.605 |
| 14 | 196 | 3.741 |
| 15 | 225 | 3.872 |
Properties of Roots
Understanding the properties of roots can help you estimate calculations mentally or verify results.
- Product Rule: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. For example, $\sqrt{16 \times 4} = \sqrt{16} \times \sqrt{4} = 4 \times 2 = 8$.
- Quotient Rule: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
- Negative Numbers: The square root of a negative number is not a real number; it is an imaginary number. For example, $\sqrt{-16} = 4i$. However, cube roots and odd roots of negative numbers are negative real numbers (e.g., $\sqrt[3]{-8} = -2$).
- Zero: The square root of zero is zero ($\sqrt{0} = 0$).
Frequently Asked Questions
What is the square root of 2?
The square root of 2 is an irrational number approximately equal to 1.41421356. It cannot be expressed as a simple fraction and its decimal representation never ends or repeats.
How do you calculate a square root manually?
The most common manual method is the "Long Division Method" or using prime factorization to simplify the radical.
For estimation, find the nearest perfect squares above and below your number to bracket the answer.
Is every positive number a square root?
Every positive number has two square roots: one positive and one negative.
For example, the square roots of 9 are +3 and -3, because both $3 \times 3$ and $(-3) \times (-3)$ equal 9.
The principal square root refers to the non-negative one.