Quick answer
The square root of n is the number x with x² = n, and radicals simplify as √(a²b) = a√b. √144 = 12, which is exact.
Enter a non-negative number. The calculator returns the square root and the simplified radical.
Related tools: Root Calculator: Nth root of a number. Cube Root Calculator: Cube root of a number. Fraction Calculator: Add, subtract, multiply and divide fractions with simplified and mixed-number results.
How to use the Square Root Calculator
- Enter number (0 or more).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Number | A number | ≥ 0 |
Formula and method
Formula
√n = x where x² = n; √(a²b) = a√b
IEEE-754 double precision; shown to at most 6 decimal places.
Floating-point square root; radical form extracts the largest square factor of whole numbers.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Number = 144.
| Result | Value |
|---|---|
| √144 | 12 |
| Simplified radical | 12 |
Example 2
Inputs: Number = 50.
| Result | Value |
|---|---|
| √50 | 7.071068 |
| Simplified radical | 5√2 |
Example 3
Inputs: Number = 2.
| Result | Value |
|---|---|
| √2 | 1.414214 |
| Simplified radical | √2 |
The first line is the headline result (√144). Below it the square Root Calculator lists simplified radical, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Simplifying radicals
√72 = √(36 × 2) = 6√2 ≈ 8.485. Factor out the largest perfect square. Radical form works for whole numbers up to a limit.
Negative numbers
Negative numbers have no real square root; √−4 = 2i in complex numbers. Perfect squares like 144 give whole-number roots, others are irrational.
Simplified radicals
√144 = 12 exactly. For non-perfect squares the calculator also gives a simplified radical: 72 = 36 × 2, so √72 = 6√2 ≈ 8.485, and √50 = 5√2 ≈ 7.071.
The square root symbol means the principal (non-negative) root. The equation x² = 9 has two solutions, 3 and −3, but √9 is only 3. Negative numbers have no real square root.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Number: entering "-4" shows "Number must be at least 0." beside that field and no result.
Choosing between the Square Root Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Root Calculator](/math/root-calculator/) | number and root | 3th root of 27 |
| [Cube Root Calculator](/math/cube-root-calculator/) | number | Cube root |
| [Fraction Calculator](/math/fraction-calculator/) | numerator 1, denominator 1 and operation | Result |
| [Ratio Calculator](/math/ratio-calculator/) | first number and second number | Simplified ratio |
| [Average Calculator](/math/average-calculator/) | numbers | Average (mean) |
Background: rounding and basic arithmetic
Rounding to a place looks at the digit just to its right: 0 to 4 rounds down and 5 to 9 rounds up (the half-up convention, so 2.5 becomes 3). Rounding 838.274 gives 800 to the nearest hundred, 840 to the nearest ten and 838 to the nearest one.
Some systems use banker's rounding (half to even), which sends 2.5 to 2 and 3.5 to 4; check which convention your task requires, and round only at the end of a multi-step calculation.
Assumptions and limitations
- Negative numbers have no real square root
- Radical form only for whole numbers up to 10^12
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for number to confirm this.
Frequently asked questions
What is √2?
√2 ≈ 1.41421, an irrational number.
What is √0.25?
√0.25 = 0.5, since 0.5 × 0.5 = 0.25.
How do I estimate a square root?
Between two perfect squares.
What is the square root of 50 in simplest form?
50 = 25 × 2, so √50 = 5√2, about 7.0711.
Why does x² = 9 have two answers when √9 has one?
The root symbol denotes only the positive root, while solving x² = 9 gives both 3 and −3.
What does the Square Root Calculator calculate?
Square root with simplified radical form. It returns √144 and Simplified radical.
What formula does the Square Root Calculator use?
√n = x where x² = n; √(a²b) = a√b. Floating-point square root; radical form extracts the largest square factor of whole numbers.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Negative numbers have no real square root.
What are the limits of this calculator?
Negative numbers have no real square root. Radical form only for whole numbers up to 10^12.
What is an example of a square Root result?
With number 144, the √144 is 12.
Related calculators
- Root Calculator — Nth root of a number.
- Cube Root Calculator — Cube root of a number.
- Fraction Calculator — Add, subtract, multiply and divide fractions with simplified and mixed-number results.
- Ratio Calculator — Simplify a ratio of two whole numbers.
- Average Calculator — Arithmetic mean, sum, count, minimum and maximum of a list of numbers.
- Long Division Calculator — Divide whole numbers to a chosen number of decimal places.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.