Square Root Calculator
Calculate the square root or any nth root of a number.
About the Square Root Calculator
Square roots and higher-degree roots (cube roots, fourth roots, and beyond) come up throughout algebra, geometry, and physics. This calculator finds the root of any degree for any positive number, defaulting to a standard square root when the degree is left at 2.
How It Works
The calculator raises the number you enter to the power of one divided by the root degree, which is mathematically equivalent to taking that root — the same identity used internally by virtually every calculator and computer.
Root = Number^(1/degree). For a square root, degree = 2; for a cube root, degree = 3.
Worked example
Scenario: Finding the square root of 144 and the cube root of −8.
- √144: 12 × 12 = 144, so the square root is 12 (a perfect square).
- ∛−8: (−2) × (−2) × (−2) = −8, so the cube root is −2.
- √2 is irrational: 1.41421356237…
Result: √144 = 12 and ∛−8 = −2.
Square roots and nth roots
The principal square root of a nonnegative number is the nonnegative value that produces the original number when squared. For example, √81 = 9. More generally, an nth root reverses raising a value to the nth power. Even roots of negative real numbers do not have real-number results; odd roots can.
Tips and common mistakes
- Even roots (2nd, 4th…) of negative numbers aren't real numbers; odd roots are.
- Root degree must be a whole number from 2 to 100.
- Estimate by bracketing: √50 is between 7 (49) and 8 (64), close to 7.07.
Assumptions & Limitations
- For even-degree roots (square root, fourth root, etc.), assumes the number entered is non-negative, since even roots of negative numbers are not real numbers.
- Odd-degree roots (like cube roots) of negative numbers are mathematically valid and handled correctly.
- Returns only the principal (positive) root for even-degree roots, not the negative counterpart, which is also technically a valid root.