Square Root Calculator
Find the principal square root, the second real root, and an exact square-back verification for any non-negative real number.
Input
Required. Use a non-negative decimal from 0 to 1 × 10³⁰⁰. U.S. thousands separators are accepted.
Relationship
y² = x ⇔ y = ±√x
Results
Answer #1 · principal square root
7.2111025509
√52 = 7.2111025509
Answer #2
– 7.2111025509
The second real value whose square is the input.
Squared check
52
Answer #1 multiplied by itself.
Verification residual
0
Absolute difference between the squared result and input.
Interpretation
Two real roots
The radical symbol √ selects the non-negative one.
Root verification
| Result | Root value | Root × root | Difference from input |
|---|---|---|---|
| Answer #1 | 7.2111025509 | 52 | 0 |
| Answer #2 | – 7.2111025509 | 52 | 0 |
Both roots square back to the same non-negative input. For zero, the positive and negative forms coincide, so there is only one distinct real root.
How to use the square root calculator
What this calculator does
This calculator evaluates the real square roots of a non-negative number. It reports Answer #1, the principal or non-negative square root written with the radical symbol √, and Answer #2, its negative counterpart. It also squares the result back to the original input and reports the numerical residual. The tool is designed for real-number arithmetic; it does not calculate imaginary roots for negative inputs, simplify symbolic radicals such as 2√3, or solve equations containing variables.
When to use it
Use it when you need a decimal approximation of a radical, when you want to check whether a number is a perfect square, when a geometry problem asks for the side length of a square from its area, or when you need to verify a manual calculation by squaring the proposed root. OpenStax explains why a positive number has two square roots while the radical sign denotes only the principal square root.
How to calculate
- The calculator opens with the ready-to-use demonstration value 52, so the results, verification table, and example Excel workbook are available immediately.
- Replace Enter your number with any valid non-negative decimal. Results update as you type; no Calculate button is required.
- Read Answer #1 for √x and Answer #2 for – √x. Check Squared check and Verification residual to confirm the calculation.
- Select Download Excel to export the current typed input and canonical results to a validated .xlsx workbook. Select Reset to clear the demonstration and all computed content. After Reset, Excel export remains unavailable until a complete valid number is entered again.
Input guide
Enter your number is the only required control. It accepts a non-negative real number from 0 through 1 × 10³⁰⁰. Plain integers, decimals such as 52.25, a leading plus sign, and correctly grouped U.S. notation such as 1,024.5 are accepted. Decimal-comma notation such as 1,5, scientific notation such as 1e3, negative values, mixed text, and malformed grouping are rejected instead of being silently reinterpreted. Increasing the input always increases the principal root, but less than proportionally: multiplying the input by four doubles its square root. A common mistake is entering a side length when the quantity under the radical should be an area or squared magnitude.
Output guide
Answer #1 is the principal square root, always zero or positive. Answer #2 is the negative real root; for an input of zero it coincides with Answer #1. Squared check is Answer #1 multiplied by itself. Verification residual is the absolute difference between that square and the input after harmless floating-point noise is normalized. The header pills repeat the current Input, Principal root, number of distinct Real roots, and whether the input is a Perfect square. The Root verification table lists each answer, its square, and its difference from the input. These are exact identities conceptually, while non-perfect-square decimal displays are rounded approximations.
Worked example
With the startup input x = 52, the calculator evaluates √52 = 7.2111025509 at the displayed precision. Therefore Answer #1 is 7.2111025509 and Answer #2 is – 7.2111025509. Squaring either value returns 52 within machine precision, so the displayed residual is 0. Because 7.2111025509 is not an integer, 52 is not a perfect square. The two answers are distinct because the input is greater than zero.
How square roots work
A square root reverses squaring. If y² = x, then y is a square root of x. For x > 0, both y and – y produce x because the product of two negative numbers is positive. The notation √x, however, is reserved for the non-negative principal value. This distinction matters when solving equations: the expression √25 equals 5, while the equation y² = 25 has the two solutions y = 5 and y = – 5.
The square-root function grows continuously for non-negative inputs, but its rate of growth slows as x increases. For example, moving from 1 to 4 increases the input by 3 and the root from 1 to 2; moving from 100 to 121 increases the input by 21 but the root only from 10 to 11. The OpenStax treatment of radicals and rational exponents connects √x with x raised to the one-half power and explains product and quotient rules.
Real-number boundary: no real number squares to a negative value. Negative radicands require complex-number arithmetic, which is outside this calculator's scope. Zero is the special boundary case: √0 = 0 and – √0 is the same number, so zero has one distinct real square root.
Accuracy, perfect squares, and manual checks
Perfect squares such as 0, 1, 4, 9, 16, 25, and 144 have integer principal roots. Most other inputs have irrational square roots whose decimal expansions never terminate or repeat, so a screen can show only an approximation. This calculator retains JavaScript's full finite numeric result for computation and export, then formats the visible value consistently. Very large or very small outputs may appear in scientific notation to remain readable.
A practical manual estimate starts by locating the input between nearby perfect squares. Because 49 < 52 < 64, it follows that 7 < √52 < 8. Repeated averaging methods can refine the estimate. For a deeper computational treatment, MIT OpenCourseWare covers square roots and Newton's method, an efficient iterative approach used in numerical computation.
When checking a result, square the proposed root rather than doubling it. Also keep the radical's scope clear: √(a + b) generally does not equal √a + √b. Finally, distinguish an approximation from an exact symbolic form. For example, √52 can be simplified exactly to 2√13, while 7.2111025509 is a decimal approximation of that same value.