Root Calculator
Calculate square roots, cube roots, and nth roots with precision. Our calculator provides step-by-step solutions, root tables, and verification of results. Perfect for mathematics, engineering, and scientific calculations.
Root Formulas and Properties
Basic Root Definitions:
• Square Root: √x = x^(1/2)
• Cube Root: ∛x = x^(1/3)
• Nth Root: ⁿ√x = x^(1/n)
Key Properties:
• (√x)² = x
• (∛x)³ = x
• (ⁿ√x)ⁿ = x
• √(a × b) = √a × √b
• ⁿ√(aⁿ) = a
• ⁿ√(1/a) = 1/ⁿ√a
Advanced Root Rules
Product Rule: ⁿ√(a × b) = ⁿ√a × ⁿ√b
Quotient Rule: ⁿ√(a ÷ b) = ⁿ√a ÷ ⁿ√b
Power Rule: ⁿ√(a^m) = (ⁿ√a)^m
Root of Root: ᵐ√(ⁿ√a) = ^(m×n)√a
Distribution: ⁿ√(aᵐ × bᵏ) = (ⁿ√a)ᵐ × (ⁿ√b)ᵏ
Special Cases:
• Even roots of negative numbers are undefined in ℝ
• Odd roots of negative numbers are negative
• ⁿ√0 = 0 for all n > 0
• ⁿ√1 = 1 for all n ≠ 0
Applications and Uses
Mathematics:
• Solving polynomial equations
• Simplifying radical expressions
• Finding geometric means
Physics:
• Wave equations and harmonics
• Quantum mechanics calculations
• Force and energy equations
Engineering:
• Structural calculations
• Signal processing
• Control systems
Other Fields:
• Financial calculations (compound interest)
• Computer graphics (vector normalization)
• Data analysis (statistical measures)