Root Calculator

Need the square root, cube root, or any nth root of a number? This calculator finds exact roots for perfect powers and decimal approximations for all other numbers.

How to Use This Calculator

1. Enter the number to find the root of (the radicand).
2. Enter the root degree (2 for square root, 3 for cube root, n for nth root).
3. Click Calculate to see the result as a decimal and, if applicable, the simplified radical form.

Root Formulas

Square root: √x = x^(1/2)

Cube root: ∛x = x^(1/3)

nth root: ⁿ√x = x^(1/n)

All roots are equivalent to fractional exponents — they share the same mathematical rules.

Perfect Squares and Cubes

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...

Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...

Example Calculations

• √144 = 12 (exact)
• √50 = √(25×2) = 5√2 ≈ 7.071
• ∛27 = 3 (exact)
• ⁴√256 = 256^(0.25) = 4
• √2 ≈ 1.41421356... (irrational)

Common Mistakes to Avoid

• √(a+b) ≠ √a + √b — A very common error. √(9+16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Always compute inside the radical first.
• Taking the square root of a negative number — Not defined in real numbers (result is imaginary: √(−4) = 2i). This calculator handles real numbers only.
• Confusing root with reciprocal — √x ≠ 1/x. √4 = 2 (not 0.25). √x = x^(1/2), not x^(−1).

Frequently Asked Questions

Why is √2 irrational?

√2 cannot be expressed as a fraction of two integers — it has infinite non-repeating decimals (1.41421356...). This was proven by the ancient Greeks and was historically groundbreaking.

How do I simplify a radical?

Factor out perfect square factors: √72 = √(36×2) = 6√2. Factor out all squares from under the radical sign and bring them outside.

What is a principal square root?

Every positive number has two square roots (+√x and −√x). The principal root is the positive one. √9 = 3 (not −3), even though (−3)² = 9 too.

Conclusion

Roots are fundamental to algebra, geometry, physics, and everyday calculations. Use this calculator to find exact or decimal values for any root — and learn to recognize perfect squares and cubes for fast mental math.

Related: Exponent Calculator | Scientific Calculator | Pythagorean Theorem Calculator