Square Root Calculator with Steps
Find any supported square root instantly. Get a decimal answer, simplified radical, perfect-square check and step-by-step explanation.
Square root calculator inputs
Quick Answer: How Do You Find a Square Root?
A square root of a number x is a value that multiplies by itself to give x. √25 = 5 because 5 × 5 = 25. The symbol √ names the principal (non-negative) square root. The equation x² = 25 still has two solutions, +5 and −5; √25 is only +5.
How to Use This Square Root Calculator
- Type a nonnegative number, a decimal, or a fraction such as 16/25.
- Press Calculate or Enter. The exact radical and a rounded decimal appear together.
- Open the steps when you want the factoring that produced the simplified radical.
- Use Advanced options only if you need a cube root, another nth root, or a different decimal length.
- Copy the exact form, the decimal, or a share link. The link is not a separate indexed page.
What Is a Square Root?
In √x, the symbol is the radical and x is the radicand. By definition, (√x)² = x when x ≥ 0. Geometry gives the same idea: √x is the side length of a square whose area is x.
√9 = 3, not ±3. People often mix that convention with solving x² = 9, which does have both +3 and −3. This calculator follows the principal-root convention used in US algebra courses.
How to Calculate Square Roots
The power form is √x = x^(1/2) for x ≥ 0. Example: 81^(1/2) = 9 because 9² = 81. For a number that is not a perfect square, the same power is an irrational decimal. √72 = 6√2, and the rounded decimal is 8.48528137 at eight places.
How to Simplify Square Roots
Pull out the largest perfect-square factor. For 72 that factor is 36:
- 72 = 36 × 2
- √72 = √36 × √2
- √72 = 6√2
- √72 ≈ 8.48528137
Stop only when nothing under the radical is still a perfect square. 2√18 is unfinished because 18 still contains 9. The finished form is 6√2. The same pairing works for √50 = 5√2 and √200 = 10√2.
Perfect Squares vs. Non-Perfect Squares
A perfect square is an integer whose square root is also an integer: 1, 4, 9, 16, 25, and so on. √144 = 12 exactly. A non-perfect square such as 50 still has an exact simplified radical (5√2) plus an unending decimal.
| n | n² | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
| 16 | 256 | 16 |
| 17 | 289 | 17 |
| 18 | 324 | 18 |
| 19 | 361 | 19 |
| 20 | 400 | 20 |
Square Root Examples
- Square root of 2: √2 = √2 ≈ 1.41421356 (irrational).
- Square root of 18: √18 = 3√2.
- Square root of 50: √50 = 5√2 ≈ 7.07106781.
- Square root of 72: √72 = 6√2 ≈ 8.48528137.
- Square root of 81: √81 = 9.
- Square root of 144: √144 = 12.
- Square root of 200: √200 = 10√2.
- Square root of 0.25: √0.25 = 1/2 because 0.25 = 1/4.
- Square root of a fraction: √(16/25) = 4/5 = 0.8.
- Negative radicand: √(-9) = 3i. There is no real square root of −9. Cube roots are different: ∛(-27) = -3.
The number box accepts a single radicand. It does not parse expressions such as √(3²+4²). Compute 9 + 16 = 25 first, then take √25.
How to Find a Square Root Without a Calculator
Place the number between neighboring perfect squares. 50 sits between 49 and 64, so √50 is a little more than 7. Newton's method refines a guess x with (x + N/x) / 2.
For N = 50 and a first guess of 7: (7 + 50/7) / 2 = (7 + 7.142857…) / 2 = 7.071428…. Squaring 7.071428 gives about 50.001, already close to 7.07106781. Repeat the same average if you need more digits.
Square Roots of Decimals and Fractions
A terminating decimal can be written as a fraction, then simplified. 0.25 = 25/100 = 1/4, so √0.25 = 1/2. For a fraction, √(a/b) = √a / √b when a ≥ 0 and b > 0. √(16/25) = 4/5 because both 16 and 25 are perfect squares.
Exact radical form is produced for integers and integer fractions whose parts are at most 100,000,000. Larger inputs still receive a decimal approximation.
Where Are Square Roots Used?
- Geometry: side of a square from its area.
- Pythagorean theorem: hypotenuse c = √(a² + b²). A 5-12-13 triangle uses √(25 + 144) = √169 = 13.
- Distance in the plane: √((x₂−x₁)² + (y₂−y₁)²).
- Statistics: standard deviation is the square root of variance. Finish that last step on the Standard Deviation Calculator.
Common Square Root Mistakes
- Reading √9 as ±3. The symbol is the principal root only.
- Writing √(a + b) = √a + √b. √(9 + 16) = 5, while √9 + √16 = 7.
- Rounding a decimal and then treating it as exact. 8.48528137 is rounded; 6√2 is exact.
- Mixing square roots with cube roots. ∛8 = 2, and odd roots of negatives are real.
- Assuming a negative radicand is “undefined” in every number system. In the complexes, √(-9) = 3i.
Frequently Asked Questions
- How do I calculate a square root?
- Use x^(1/2) for x ≥ 0, or factor out a perfect square. √25 = 5 because 5² = 25.
- How do I simplify a square root?
- Factor the radicand as (perfect square) × (rest). √72 = 6√2 from 36 × 2.
- What is the square root of 72?
- √72 = 6√2 ≈ 8.48528137 at eight decimal places.
- What is the square root of 50?
- √50 = 5√2 ≈ 7.07106781.
- What is the square root of 144?
- √144 = 12 because 12 × 12 = 144.
- How do I find the square root of a decimal?
- Write the decimal as a fraction when it terminates. √0.25 = 1/2. Otherwise use the decimal approximation.
- How do you find the square root of a fraction?
- √(a/b) = √a / √b for a ≥ 0 and b > 0. √(16/25) = 4/5.
- What is a perfect square?
- An integer that equals some integer times itself. 81 is a perfect square because √81 = 9.
- Can a negative number have a square root?
- Not in the real numbers. The principal complex value is 3i for −9.
- What is the difference between square root and cube root?
- A square root solves y² = x. A cube root solves y³ = x and can be negative: ∛(-27) = -3. Open Advanced options to switch degree on this page. There is no separate cube-root URL on EverydayTools.
- Does every positive number have two square roots?
- Yes, +√x and −√x. The calculator reports the principal (plus) root. Zero has only one square root.
- How do I calculate square roots without a calculator?
- Estimate between perfect squares, then average x and N/x. For 50, one step from 7 already reaches about 7.07142857.
Methodology and Limits
Square roots use the browser's IEEE-754 square-root for the decimal and integer factoring for the exact radical. Intermediates stay unrounded. The displayed decimal uses fixed-place rounding (default 8 places) and is never labeled exact. Nth roots use x^(1/n), with odd-degree negatives computed from the absolute value and then signed. Complex results are implemented for square roots only: √(−a) = √a · i.
Exact simplify runs for integer parts through 100,000,000 so the page cannot freeze on huge factorization. Inputs whose absolute value exceeds 1,000,000,000,000 are rejected. Share URLs may contain the number you typed; they canonicalize to https://everydaytools.pro/square-root-calculator. Nothing is uploaded.