Square Root Calculator
Calculate square roots, cube roots, and nth roots of any number with step-by-step solutions.
About Square Root Calculator
A square root calculator finds the square root, cube root, or nth root of any positive number. Square roots are fundamental in geometry (finding side lengths), physics (wave equations), statistics (standard deviation), finance (volatility calculations), and many other fields. The square root of a number x is a value that, when multiplied by itself, equals x. For example, the square root of 25 is 5, because 5 × 5 = 25.
How to Use
Enter the number you want to find the root of, then select the root type: square root (n=2), cube root (n=3), or any custom nth root. The calculator instantly displays the result with full decimal precision. For square roots of perfect squares, it shows the exact integer answer. For non-perfect squares, it provides both the simplified radical form (if applicable) and the decimal approximation.
Formula / Key Equations
Square root: x^(1/2) or sqrt(x). Cube root: x^(1/3) or cbrt(x). Nth root: x^(1/n). For example: sqrt(144) = 12, cbrt(27) = 3, 4th root of 16 = 2. Negative numbers have real nth roots only when n is odd: cbrt(-8) = -2, but sqrt(-4) is undefined in real numbers (it equals 2i in complex numbers).
Common Use Cases
Finding side lengths of squares and diagonals of rectangles. Calculating standard deviation in statistics. Computing distance formulas (Euclidean distance between two points). Solving quadratic equations using the quadratic formula. Calculating electrical impedance in AC circuits. Working with the Pythagorean theorem in geometry.
Limitations
The calculator works with real numbers only — complex roots of negative numbers are not supported. For very large numbers, floating-point precision limits may affect the last few decimal places. Perfect square detection works for integers up to very large values. Simplified radical form is shown for square roots of numbers with square factors.
Frequently Asked Questions
What is the square root of a negative number?
In real numbers, the square root of a negative number is undefined. In complex numbers, the square root of -1 is defined as i (the imaginary unit). For example, sqrt(-4) = 2i. Our calculator works with real numbers only and will indicate if a result is not a real number.
How do I calculate the square root without a calculator?
Common manual methods include: prime factorization (for perfect squares), the Babylonian/Herons method (iterative approximation), and using a number line estimation. For example, to estimate sqrt(50): 7 squared is 49 and 8 squared is 64, so sqrt(50) is approximately 7.07.
What is the difference between square root and cube root?
The square root of x is a number that multiplied by itself twice equals x (n=2). The cube root of x is a number that multiplied by itself three times equals x (n=3). For example, sqrt(8) = 2.83 (since 2.83 squared equals 8), while cbrt(8) = 2 (since 2 cubed equals 8).
What is a simplified radical form?
A simplified radical has no perfect square factors inside the radical sign. For example, sqrt(72) = sqrt(36 × 2) = 6×sqrt(2). The calculator shows this simplified form when applicable, which is preferred in mathematical work and answers.
Can I calculate roots of fractional numbers?
Yes. The calculator handles fractional inputs. For example, the square root of 0.25 is 0.5, and the cube root of 1/8 is 0.5. You can enter decimals or fractions as the input number.
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