Square Root Calculator
Find the square root of any number to as many as 20 decimal places. The calculator tells you whether the root is exact, gives the simplified radical form of whole numbers (72 becomes 6√2) and handles negative numbers as imaginary roots.
A negative number gives an imaginary root.
Result
Enter a number to see its square root.
The calculation runs in this browser tab. Nothing you enter is uploaded, saved or logged.
What is a square root?
The square root of a number n is the number that gives n when multiplied by itself. The square root of 49 is 7, because 7 × 7 = 49. The root sign is √, so √49 = 7.
Every positive number has two square roots, one positive and one negative (7 and −7 both square to 49). The √ symbol means the non-negative one, called the principal square root, and that is the value this calculator returns. Numbers such as 16, 2.25 or 144 are perfect squares and have an exact root; most others, like 2 or 50, have a root that is irrational, with decimals that never end and never repeat, so any decimal shown for them is an approximation.
How to use the Square Root Calculator
- Type the number whose root you want. Decimals and negative numbers are accepted.
- Choose how many decimal places you want from 2 to 20.
- Read the root. A leading ≈ means the value is rounded; no ≈ means it is exact.
- For whole numbers, the details also show the simplified radical form and whether the number is a perfect square. Copy result copies the plain number.
Formula
- Definition
- √n = x, where x ≥ 0 and x × x = n
- Simplified radical form
- √n = a√b, where a² is the largest square dividing n and b = n ÷ a²
- Negative numbers
- √(−n) = i × √n, where i² = −1
The digits are computed with whole-number arithmetic: the calculator finds the largest integer whose square does not exceed n × 10^(2d), then rounds correctly to d decimals. This does not rely on floating point, so all 20 decimals are right.
Simplifying a radical means pulling out perfect-square factors: 72 = 36 × 2, so √72 = √36 × √2 = 6√2.
Worked example
The square root of 50 to 6 decimals, with its simplified form. These are the results the calculator gives:
| Question | Result |
|---|---|
| Input: Number | 50 |
| Input: Decimal places | 6 decimal places |
| Square root of 50 | ≈ 7.071068 |
| Exact value | No, the square root is irrational or longer than 6 decimals; rounded |
| Simplified radical form | 5√2 |
| Perfect square | No |
Assumptions and limits
- Results are rounded half up at the last decimal shown and marked with ≈ unless the root is exact within the decimals you chose.
- A root with more decimals than you asked for is shown rounded and marked ≈ even when it is rational: √0.00000001 is exactly 0.0001, but at 2 decimals it appears as ≈ 0.00.
- The radical form is given for whole numbers up to 10^12. Larger numbers and decimals show only the decimal root.
- The root of a negative number is shown as a multiple of i (the imaginary unit). The calculator does not work with other complex numbers.
- Inputs can have up to 30 digits. Use a dot for decimals and no thousands separators.
Frequently asked questions
How do I calculate a square root by hand?
Find the perfect squares on either side of the number, estimate between them, then refine. For √50: 7² = 49 and 8² = 64, so the root is just above 7. Averaging 7 with 50 ÷ 7 gives about 7.07, and repeating the step converges quickly to 7.0710678…
What is the square root of a negative number?
No real number squared gives a negative result, so the root of a negative number is imaginary. By definition i² = −1, which gives √(−4) = 2i. The calculator shows the principal root as a multiple of i.
How do I know if a number is a perfect square?
A perfect square is the square of a whole number, so its root is a whole number. The calculator lists “Perfect square: Yes” with the root when that is the case, for example 144 = 12². A decimal such as 2.25 also has an exact root (1.5) even though it is not a whole-number square.
What does the simplified radical form mean?
It writes the root with the smallest possible number under the root sign. 50 = 25 × 2, so √50 = 5√2. This is the exact value; the decimal 7.0710678… is only an approximation of it.
Is √2 a rational number?
No. √2 is irrational: it cannot be written as a fraction of whole numbers, and its decimals continue forever without repeating. Any decimal for it, such as 1.41421356237, is rounded.
Why does the result sometimes end with ≈ and sometimes not?
Without ≈ the value shown is the exact root. With ≈ the true root has more decimals than are displayed, so the figure was rounded to the number of places you chose.