Radical Calculator
Calculate nth roots, simplify radicals, and solve radical equations
Radical Calculator
The number under the radical sign
The degree of the root (2 = square root, 3 = cube root, etc.)
Calculation Results
Step-by-Step Solution:
Decimal Approximation:
Verification:
Radical Examples
Square Root
Problem: √16
Solution: 4
Why: Because 4² = 16
Cube Root
Problem: ∛27
Solution: 3
Why: Because 3³ = 27
Fourth Root
Problem: ⁴√81
Solution: 3
Why: Because 3⁴ = 81
Simplified Radical
Problem: √50
Solution: 5√2
Why: Because 50 = 25 × 2 = 5² × 2
Radical Operations
Adding/Subtracting Radicals
Only radicals with the same degree and radicand can be added/subtracted:
Multiplying Radicals
Multiply radicals with the same degree:
Dividing Radicals
Divide radicals with the same degree:
Common Radicals
Quick Examples
Radical Properties
ⁿ√(aⁿ) = a for a ≥ 0
ⁿ√(a × b) = ⁿ√a × ⁿ√b
ⁿ√(a ÷ b) = ⁿ√a ÷ ⁿ√b
ⁿ√x = x^(1/n)
Understanding Radicals
What is a Radical?
A radical is a mathematical symbol (√) used to indicate roots of numbers. The radical sign with a number indicates which root to find. For example, ³√8 means "find the number that when cubed equals 8."
Radical Notation
ⁿ√x = x^(1/n)
n = radical degree, x = radicand
Key Components
- •Radical symbol (√): The root sign
- •Index (n): The degree of the root
- •Radicand (x): The number under the radical
Types of Roots
Important Rules
- Even roots of negative numbers are not real
- Odd roots of negative numbers are negative
- √0 = 0 for any positive degree
- √1 = 1 for any positive degree
Applications of Radicals
Geometry
Used to find side lengths in right triangles, areas of circles, and distances between points.
Physics
Appears in formulas for velocity, acceleration, wave mechanics, and quantum physics calculations.
Finance
Used in compound interest calculations, standard deviation, and risk assessment models.