Simplifying Radicals Calculator
Simplify square roots, cube roots, and higher-order radicals with step-by-step solutions
Simplify Radical Expressions
Number in front of radical
Number under the radical sign
2 = square root, 3 = cube root
Simplified Radical
Common Examples
√72 = 6√2
2√50 = 10√2
∛16 = 2∛2
√288 = 12√2
4√32 = 16√2
⁴√81 = 3
Simplification Rules
Factor the Radicand
Find the prime factorization of the number under the radical
Extract Perfect Powers
Remove groups of factors equal to the root index
Simplify the Result
Combine coefficients and reduce the index if possible
Quick Tips
√(a²) = |a| for all real numbers
√(ab) = √a × √b when a,b ≥ 0
ⁿ√(aⁿ) = a for odd n, |a| for even n
Only add/subtract like radicals
Rationalize denominators when possible
Understanding Radical Simplification
What are Radicals?
Radicals are expressions that contain a root symbol (√ or ⁿ√). The number under the radical sign is called the radicand, and the small number indicating the root is called the index (2 for square roots, 3 for cube roots, etc.).
Why Simplify Radicals?
- •Makes calculations easier and more accurate
- •Provides the standard mathematical form
- •Helps identify patterns and relationships
- •Required for many algebraic operations
Simplification Process
Example: Simplify √72
Step 1: Factor 72 = 2³ × 3² = 8 × 9
Step 2: √72 = √(36 × 2) = √36 × √2
Step 3: √72 = 6√2
Example: Simplify ∛54
Step 1: Factor 54 = 2 × 3³ = 2 × 27
Step 2: ∛54 = ∛(27 × 2) = ∛27 × ∛2
Step 3: ∛54 = 3∛2