Square Root Calculator with Steps
Find the principal square root of an integer, decimal or fraction. See a decimal approximation, exact or simplified radical form, perfect-square status, equation solutions, nearby squares and clear calculation steps. Negative inputs return pure-imaginary roots.
Last Updated: July 25, 2026
Online Square Root Calculator
Enter one value. Valid examples include 144, 72, 0.81, 2.5e4, 9/16 and -81.
Your exactness and rounding status will appear here.
Calculation steps
- Your square-root steps will appear here.
Try an example
Recent calculations
- Your last eight results will appear here.
How to Use This Square Root Calculator
- Enter one signed integer, decimal, scientific-notation value or fraction. Examples include
144,0.81,2.5e4,9/16and-81. - Select Auto precision or a fixed number of decimal places. Auto keeps an exact terminating answer when possible and shows an irrational result to as many as 12 decimal places.
- Select Calculate Square Root or press Enter.
- Read the principal root, exact or simplified form, number type, nearby integer squares and calculation steps.
- Use Copy Result to copy the visible decimal answer. Select an example or a recent calculation to load it again.
The calculator accepts one slash for a fraction, but it does not accept commas, units, percent signs, radical symbols or complete expressions. Enter 25000, not 25,000. Enter 9/16, not √(9/16). A negative input has no real root, so the tool reports its principal pure-imaginary root using i.
What Is a Square Root?
A square root reverses squaring. If a non-negative number y satisfies y² = x, then y is the principal square root of x. The radical sign √ identifies the operation, and the value under the sign is the radicand. For example, √49 = 7 because 7 × 7 = 49.
The radical symbol returns the principal, non-negative root. This point prevents a common sign error. √49 equals 7, not ±7. The equation x² = 49 has two solutions, x = 7 and x = −7, because both values produce 49 when squared. A square-root expression and a quadratic equation answer different questions.
Zero and one are useful boundary cases. √0 = 0 because 0² = 0. √1 = 1 because 1² = 1. In the real-number system, √x is defined only for x ≥ 0. For a negative real input, the calculator uses i² = −1 and reports a pure-imaginary principal root. For example, √(−81) = 9i.
Square Root Formula
√(−x) = i√x, where x > 0 and i² = −1
Verification: y² = x
The exponent form x^(1/2) and the radical form √x express the same operation for a non-negative real value. The calculator reduces a fraction before testing whether its numerator and denominator are both perfect squares. It also checks integer radicands for a square factor that can move outside the radical.
A perfect square has an integer square root. Examples include 0, 1, 4, 9, 16, 25 and 36. A positive rational value also has an exact rational square root when its reduced numerator and denominator are perfect squares. For example, √(9/16) = 3/4. If either part is not a square, the result is irrational.
Understanding the Calculator Results
| Result field | What it means | Example for 72 |
|---|---|---|
| Principal square root | The non-negative decimal root at your selected precision. | 8.485281374239 |
| Exact or simplified form | An integer, rational value or radical with square factors removed. | 6√2 |
| Number type | Perfect square, exact rational root or irrational root. | Irrational square root |
| Nearby integer squares | The consecutive whole-number squares that bound the radicand. | 8² = 64 < 72 < 81 = 9² |
| Equation solutions | The positive and negative values that solve x² = input. | x = ±6√2 |
| Calculation steps | The reduction, factorization, approximation and verification path. | 72 = 36 × 2 |
A displayed decimal may be rounded even when the exact radical stays unchanged. For √2, the exact answer is √2, while 1.414213562373 is a decimal approximation. Do not replace the exact form with a rounded decimal too early when later work requires precision.
How to Simplify Square Roots
To simplify an integer radical, find the largest perfect-square factor of the radicand. Split the radical into a product, take the square root of the perfect-square factor and leave the remaining factor under the radical.
√72 = √(36 × 2) = √36 × √2 = 6√2
The form 6√2 is simplified because 2 has no perfect-square factor greater than 1. Squaring the result verifies it: (6√2)² = 36 × 2 = 72. The same process works through prime factorization. Pair equal prime factors, move one factor from each pair outside the radical and keep every unpaired factor inside.
Not every radical changes. √7 is already simplified because 7 is prime. √45 becomes 3√5 because 45 = 9 × 5. √200 becomes 10√2 because 200 = 100 × 2. For large integers beyond the tool’s simplification limit, the calculator still returns a decimal approximation and an unsimplified exact radical.
Worked Square Root Examples
Example 1: Perfect square
Find √144. Since 12 × 12 = 144, the principal root is exactly 12. The nearby-square check collapses to 12² = 144. If solving x² = 144, the equation solutions are x = ±12, but the radical itself remains 12.
Example 2: Simplified radical
Find √72. The largest square factor is 36. Rewrite 72 as 36 × 2, apply the product rule and calculate √36 = 6. The exact result is 6√2. Its decimal approximation is about 8.485281374239, which lies between 8 and 9 because 64 < 72 < 81.
Example 3: Decimal
Find √0.81. Write 0.81 as 81/100. Both numerator and denominator are perfect squares, so √(81/100) = 9/10 = 0.9. This is an exact terminating decimal, not a rounded estimate.
Example 4: Fraction
Find √(9/16). Reduce the fraction if needed, then take the square root of each part: √9/√16 = 3/4 = 0.75. A fraction such as 2/3 does not produce a rational root because 2 and 3 are not perfect squares.
Example 5: Irrational root
Find √2. No rational number squared equals 2, so the exact form remains √2. A numerical method gives 1.414213562373095… . At six decimal places, standard nearest rounding produces 1.414214.
Example 6: Negative input
Find √(−81). No real number squared equals −81. Since √81 = 9 and √(−1) = i, the principal complex root is 9i. The equation x² = −81 has two solutions, x = ±9i.
How Decimal Approximation Works
For a non-perfect square, the decimal digits continue without ending or repeating in a fixed block. The calculator uses the browser’s square-root operation for the approximation and keeps the exact radical separately. Auto precision shows up to 12 places for a typical irrational result. A fixed choice shows the requested number of places and applies nearest rounding.
One manual approach is Newton’s method. Start with a positive estimate g and repeatedly average g with x/g. Each suitable iteration moves the estimate closer to √x.
For x = 2 and g = 1.5, the next estimate is (1.5 + 2/1.5)/2 = 1.416666…. Repeating the process quickly approaches 1.41421356237. The result steps identify exact roots and simplified radicals before showing the numerical approximation.
Perfect Squares from 0 to 20
| n | n² | √(n²) | n | n² | √(n²) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 11 | 121 | 11 |
| 1 | 1 | 1 | 12 | 144 | 12 |
| 2 | 4 | 2 | 13 | 169 | 13 |
| 3 | 9 | 3 | 14 | 196 | 14 |
| 4 | 16 | 4 | 15 | 225 | 15 |
| 5 | 25 | 5 | 16 | 256 | 16 |
| 6 | 36 | 6 | 17 | 289 | 17 |
| 7 | 49 | 7 | 18 | 324 | 18 |
| 8 | 64 | 8 | 19 | 361 | 19 |
| 9 | 81 | 9 | 20 | 400 | 20 |
| 10 | 100 | 10 |
These values help estimate an unknown root. Since 100 < 110 < 121, √110 lies between 10 and 11. Comparing distances suggests it is closer to 10, while a numerical calculation gives about 10.4880884817.
Common Uses of Square Roots
- Square dimensions: a square with area A has side length √A.
- Right triangles: the Pythagorean theorem gives c = √(a² + b²) for the hypotenuse.
- Coordinate distance: distance uses the square root of the sum of squared coordinate differences.
- Statistics: standard deviation is the non-negative square root of variance.
- Physics and engineering: many magnitude, speed and root-mean-square formulas include square roots.
- Scaling: when area changes by a factor, a matching linear dimension changes by the square root of that factor.
Keep measurement units consistent before applying a formula. If an area is stated in square metres, its square-root result is measured in metres. The calculator processes the number only and does not infer or convert units.
Common Square Root Mistakes
- Writing ± for a principal radical. √25 is 5; ±5 belongs to the solutions of x² = 25.
- Assuming √(a + b) = √a + √b. This identity is generally false.
- Rounding an irrational root during an intermediate step and carrying the error forward.
- Leaving a square factor inside a radical, such as writing √72 instead of the simpler 6√2.
- Treating an imaginary result as real. A negative input produces a pure-imaginary root in this tool.
- Forgetting to reduce a fraction before checking whether its numerator and denominator are squares.
Accuracy, Privacy and Limits
The input parser reduces finite decimals and fractions with integer arithmetic, checks exact rational squares and simplifies supported integer radicals. Decimal approximations use the browser’s standard Math.sqrt method, which follows normal double-precision floating-point limits. The display supports up to 15 decimal places, but source-data accuracy still controls the meaning of a real-world answer.
Inputs may contain up to 80 significant digits with a controlled exponent range. Radical simplification is limited to manageable integers so the page stays responsive. Your input, answer and recent history remain in the current browser page and disappear when it reloads. The tool does not execute entered text as code or send the entered number to a calculation server.
Square Root Calculator vs Other Math Tools
| Tool | Best use | Main output |
|---|---|---|
| Square Root Calculator | One radicand, simplification and root-specific steps | Principal root, exact radical, type, bounds and equation solutions |
| Math Calculator | Complete arithmetic expressions and percentages | General numerical result |
| Scientific Calculator | Trigonometry, logarithms, powers and advanced functions | Advanced expression result |
| Fraction Calculator | Fraction operations and mixed-number conversion | Simplified rational result |
| Average Calculator | Summarizing a list of observations | Mean, sum, count and range |
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Frequently Asked Questions
What is a square root?
A square root of x is a number that produces x when multiplied by itself. The symbol √x means the principal, non-negative square root.
How do I use the square root calculator?
Enter a signed integer, decimal, scientific-notation value or fraction, select the decimal precision and choose Calculate Square Root. The result includes the principal root, exact or simplified form, number type, equation solutions, bounds and steps.
What is the principal square root?
The principal square root is the non-negative root represented by the radical symbol. For example, the principal square root of 36 is 6.
Why does √25 equal 5 instead of ±5?
The radical symbol returns the principal non-negative root, so √25 = 5. The equation x² = 25 has two solutions, x = 5 and x = −5.
What is a perfect square?
A perfect square is an integer equal to another integer multiplied by itself. Examples include 0, 1, 4, 9, 16, 25, 36 and 49.
How do you simplify a square root?
Find the largest perfect-square factor, split the radicand into a product and move the square root of that factor outside the radical. For example, √72 = √(36 × 2) = 6√2.
Can the calculator find the square root of a decimal?
Yes. Enter a non-negative decimal such as 0.81. The calculator reduces it to an exact fraction when possible, then shows its exact or approximate root.
Can a negative number have a real square root?
No. A real number squared cannot be negative. This calculator reports the principal pure-imaginary root instead. For example, √(−81) = 9i.
What are the square roots of 0 and 1?
√0 = 0 and √1 = 1. Both are perfect squares and exact boundary cases in the non-negative real-number domain.
Why do some square roots need decimal approximations?
A positive rational value that is not the square of a rational number has an irrational square root. Its decimal digits do not terminate or repeat, so the display must round them.
What is the difference between squaring and finding a square root?
Squaring multiplies a number by itself. Finding a square root reverses that operation by finding the non-negative value whose square equals the radicand.
How do you estimate a square root without a calculator?
Place the radicand between two nearby perfect squares, estimate a value between their roots and refine it. Newton’s method refines a guess g with (g + x/g) ÷ 2.