Root Calculator
Calculate nth roots of numbers with step-by-step solutions and radical simplification
Calculate Root
Calculate the nth root of a number:
The degree of the root (2 = square root, 3 = cube root, etc.)
The number under the radical sign
Result
✓ This is a perfect 2$nd root!
Step-by-Step Solution
Calculate the 2nd root of 64
Prime factorization of 64: 2 × 2 × 2 × 2 × 2 × 2
Simplified form: 8
Decimal approximation: 8.000000
Common Root Examples
Square Roots
√4 = 2 (perfect square)
√9 = 3 (perfect square)
√16 = 4 (perfect square)
√2 ≈ 1.414
√3 ≈ 1.732
Cube Roots
³√8 = 2 (perfect cube)
³√27 = 3 (perfect cube)
³√64 = 4 (perfect cube)
³√-8 = -2 (odd roots of negatives)
³√2 ≈ 1.260
Types of Roots
Square Root
√a = b where b² = a
Most common root
Cube Root
³√a = b where b³ = a
Works with negative numbers
nth Root
ⁿ√a = b where bⁿ = a
General form for any index
Quick Reference
Square root symbol (index = 2)
Cube root symbol (index = 3)
nth root symbol (any index)
Radicand: number under radical
Index: degree of the root
Even roots of negatives = complex
Understanding Roots in Mathematics
What is a Root?
A root is the inverse operation of exponentiation. The nth root of a number a is a number b such that b^n = a. In other words, roots help us find what number, when raised to a certain power, gives us the original number.
Root Formula
ⁿ√a = b ⟺ bⁿ = a
- n: The index (degree) of the root
- a: The radicand (number under the radical)
- b: The result of the root operation
Properties of Roots
- •ⁿ√(ab) = ⁿ√a × ⁿ√b (product property)
- •ⁿ√(a/b) = ⁿ√a / ⁿ√b (quotient property)
- •(ⁿ√a)^n = a (inverse property)
- •ⁿ√aᵐ = a^(m/n) (power property)
Special Cases
- Perfect roots: Results are integers (√16 = 4)
- Odd roots: Can be negative (³√-8 = -2)
- Even roots: Negative radicands give complex results
- Fractional indices: Express roots as exponents