Root Calculator
Calculate square roots, cube roots, and nth roots with this free online calculator. Get precise results for any root calculation.
Square Root Calculator
Cube Root Calculator
General Root Calculator
About Root Calculations
A root of a number is a value that, when multiplied by itself a certain number of times, gives the original number. Roots are the inverse operation of exponentiation.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 × 3 = 9.
√16 = 4 because 4² = 16
√2 ≈ 1.414 (irrational number)
Cube Root
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8.
∛27 = 3 because 3³ = 27
∛-8 = -2 (cube roots can be negative)
Nth Root
The nth root of a number is a value that, when raised to the power of n, gives the original number. For example, the 4th root of 16 is 2, because 2⁴ = 16.
⁵√32 = 2 because 2⁵ = 32
Properties of Roots
| Property | Formula | Example |
|---|---|---|
| Product Rule | ⁿ√(a × b) = ⁿ√a × ⁿ√b | √(4 × 9) = √4 × √9 = 2 × 3 = 6 |
| Quotient Rule | ⁿ√(a ÷ b) = ⁿ√a ÷ ⁿ√b | √(16 ÷ 4) = √16 ÷ √4 = 4 ÷ 2 = 2 |
| Power Rule | ⁿ√(a^m) = (ⁿ√a)^m | √(9²) = (√9)² = 3² = 9 |
| Root of Root | ᵐ√(ⁿ√a) = ᵐⁿ√a | √(∛8) = ⁶√8 = ⁶√2³ = 2^(3/6) = 2^(1/2) = √2 |
Applications of Roots
- Geometry: Finding the length of a side in a right triangle (Pythagorean theorem)
- Physics: Calculating the period of a pendulum
- Engineering: Determining structural stability
- Finance: Calculating compound interest rates
- Computer science: Algorithms for data compression and encryption