Square Root Calculator
Calculate the square root, cube root, or any nth root of a number instantly.
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The square root calculator finds the square root, cube root, or any nth root of a number instantly. Enter the number and the root degree — 2 for square root, 3 for cube root, or any value for higher roots — and get a precise result to six decimal places. Square roots come up constantly in practical contexts: calculating the side length of a square from its area, finding the diagonal of a rectangle using the Pythagorean theorem, working out standard deviation in statistics, and solving quadratic equations in algebra and physics. The cube root appears in volume problems — if a cube has a volume of 1,000 cm³, the side length is the cube root of 1,000 = 10 cm. For higher-order roots, the calculator handles any positive degree. Engineers use 4th and 5th roots in certain signal processing formulas; scientists encounter nth roots in decay and growth rate calculations. The root degree field accepts decimals too, which makes it equivalent to a fractional exponent calculator.
How to Use the Square Root Calculator
The Square Root Calculator is designed to give you an accurate answer in seconds. Follow these steps:
- Step 1: Enter your number in the Number field. The minimum value is 0. The default is 144. Adjust this to match your specific situation.
- Step 2: Enter your root degree (2 = square root, 3 = cube root) in the Root Degree (2 = square root, 3 = cube root) field. The minimum value is 1. The default is 2. Adjust this to match your specific situation.
- Step 3: Click Calculate to see your results instantly. The output updates as soon as you submit.
No account or sign-up required. All calculations run locally in your browser — nothing is stored or transmitted to any server.
Example Calculation
Here is what the Square Root Calculator produces with its default values. Change any input above to recalculate instantly for your own figures.
Inputs
- Number144
- Root Degree (2 = square root, 3 = cube root)2
Results
- Result12.000000
- Square Root (√)12.000000
- Cube Root (∛)5.241483
How It Works
nth Root of x = x^(1/n)
Formula: nth Root of x = x^(1/n) Every root is a fractional exponent. The square root of x is x^(1/2), the cube root is x^(1/3), the 4th root is x^(1/4), and so on. A root is the inverse of raising to a power. Examples: √144 = 144^(1/2) = 12 (because 12² = 144) ∛27 = 27^(1/3) = 3 (because 3³ = 27) 4th root of 625 = 625^(1/4) = 5 (because 5⁴ = 625) For negative numbers, even roots (√, 4th root, etc.) have no real solution — you cannot square a real number and get a negative result. Odd roots (∛, 5th root, etc.) of negative numbers are valid: the cube root of −8 is −2, because (−2)³ = −8. Most square roots produce irrational numbers — they cannot be expressed as exact fractions. √2 = 1.41421356..., √3 = 1.73205080..., √5 = 2.23606797..., continuing infinitely without repeating. The calculator shows these to six decimal places, which is precise enough for virtually any practical or academic use.
Frequently Asked Questions
What is the square root of 144?
The square root of 144 is 12. This is a perfect square — 12 × 12 = 144. Other common perfect squares: √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √169 = 13, √196 = 14, √225 = 15. Numbers that are not perfect squares produce irrational results with infinitely many decimal places.
How do I calculate a square root without a calculator?
For perfect squares, memorise the common ones up to 15² = 225. For other numbers, use the estimation method: find the two consecutive integers the root falls between (√50 is between 7 and 8 since 7² = 49 and 8² = 64), then narrow in with trial multiplication. The Babylonian method (repeatedly averaging your guess with number/guess) converges quickly for any number.
What is the difference between square root and cube root?
The square root asks "what number multiplied by itself gives this value?" The cube root asks "what number multiplied by itself three times gives this value?" √64 = 8 because 8² = 64. ∛64 = 4 because 4³ = 64. Square roots are used for area and 2D geometry; cube roots appear in volume and 3D problems.
Can you take the square root of a negative number?
Not in the real number system — no real number squared gives a negative result. The square root of a negative number produces an imaginary number, written with i (the imaginary unit, where i² = −1). √−9 = 3i. Imaginary and complex numbers are used in electrical engineering, quantum physics, and signal processing, but are outside the scope of this calculator.
Is the square root calculator free?
Yes — free with no sign-up needed. Handles square roots, cube roots, and any nth root. All calculations run in your browser and no data is stored.