Root Calculators
Square roots, cube roots, nth roots, and quadratic equation solver — all in one place
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√ Square Root
Finds what number multiplied by itself equals your input. Used for calculating distances, areas, and standard deviation in statistics.
³√ Cube Root
Finds what number multiplied by itself three times equals your input. Essential for volume calculations and physics problems.
ⁿ√ Nth Root
General root calculator for any degree. Useful for compound interest, population growth, and scientific formulas.
📐 Root Calculation Formulas
Square Root
Example: √144 = 12 because 12 × 12 = 144
Cube Root
Example: ³√27 = 3 because 3 × 3 × 3 = 27
Note: Cube roots work with negative numbers too. ³√-8 = -2 because (-2)³ = -8.
Nth Root
Example: ⁴√16 = 2 because 2⁴ = 16
For even roots (2nd, 4th, 6th...), the number must be positive. For odd roots, negatives are allowed.
Quadratic Formula
The discriminant (b² - 4ac) tells you how many real roots exist:
• Positive → Two real roots
• Zero → One real root
• Negative → No real roots (complex numbers)
🎯 Real-World Examples
You need to find the diagonal of a square room that is 15 feet wide. Diagonal = √(15² + 15²) = √450 ≈ 21.2 feet.
A shipping container has volume 125 cubic meters. Side length = ³√125 = 5 meters.
An investment grows from $1,000 to $2,000 in 5 years. Annual growth rate = ⁵√(2000/1000) - 1 ≈ 14.9%.
A ball thrown upward: h = -5t² + 20t + 2. Solve -5t² + 20t + 2 = 0 to find when it hits the ground.
❓ Frequently Asked Questions
Square root (√) finds what number multiplied by ITSELF equals the original. Cube root (³√) finds what number multiplied by ITSELF THREE TIMES equals the original.
No — square roots of negative numbers are not real numbers (they are imaginary numbers). Our calculator will show "Not a real number".
Yes! Cube roots of negative numbers are real. For example, ³√-8 = -2 because (-2) × (-2) × (-2) = -8.
Nth root means any root degree. 2nd root = square root, 3rd = cube root, 4th = fourth root, and so on.
In quadratic equations, the discriminant (b² - 4ac) tells you how many real roots exist. Positive = two, zero = one, negative = none (complex).
Results are precise to 6 decimal places, using JavaScript's 64-bit floating point math (about 15 digits of precision).
