Radicals Calculator - Calculate Nth Roots
Result:
Welcome to our Radicals Calculator. This tool calculates the nth root of any number, allowing you to work with various radical expressions and root indices.
What are Radicals?
Radicals are expressions that contain roots. The radical symbol √ indicates finding a root, with the index determining which root to find.
Understanding Radicals
A radical expression has three parts:
- Radical symbol (√): Indicates a root operation
- Index (n): The type of root (square, cube, etc.)
- Radicand: The number under the radical symbol
Common Radicals
√x (n=2): Square root
∛x (n=3): Cube root
⁴√x (n=4): Fourth root
⁵√x (n=5): Fifth root
ⁿ√x: Nth root (general form)
Properties of Radicals
Product rule: ⁿ√(ab) = ⁿ√a × ⁿ√b
Quotient rule: ⁿ√(a/b) = ⁿ√a / ⁿ√b
Power rule: ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Rational exponent: ⁿ√x = x^(1/n)
Examples of Radicals
Square Root
√16 = 4
Index = 2 (implied)Cube Root
∛27 = 3
Index = 3Fourth Root
⁴√81 = 3
Index = 4Even vs Odd Roots
Even Roots (n = 2, 4, 6, ...)
Domain: Non-negative real numbers only
Principal root: Always positive
Example: √9 = 3 (not ±3)
Restriction: ⁿ√(-x) is undefined for real numbers
Odd Roots (n = 3, 5, 7, ...)
Domain: All real numbers
Sign preservation: Same sign as radicand
Example: ∛(-8) = -2
Property: ⁿ√(-x) = -ⁿ√x
Applications of Radicals
- Geometry: Finding side lengths in right triangles
- Physics: Calculating velocities and accelerations
- Engineering: Signal processing and wave analysis
- Finance: Compound interest calculations
- Statistics: Standard deviation computations
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