Multiplying Radicals Calculator
Multiply radical expressions with different indices and simplify results step-by-step
Multiply Radical Expressions
First Radical
Number outside the radical
Number under the radical
Root type (2 = √, 3 = ∛, etc.)
Expression: √0
Second Radical
Number outside the radical
Number under the radical
Root type (2 = √, 3 = ∛, etc.)
Expression: √0
Quick Examples
Square Roots
3√12 × 5√18 = 15√216 = 90√6
Mixed Indices
2∛12 × 5√18 = 10⁶√(12² × 18³)
Perfect Powers
√16 × √25 = √400 = 20
Multiplication Rules
Same Index: ᵃ√b × ᵃ√c = ᵃ√(b×c)
Coefficients: Multiply numbers outside separately
Different Indices: Convert to common index using LCM
Simplify: Extract perfect powers from radicand
Understanding Radical Multiplication
Basic Principles
When multiplying radicals, we follow specific rules based on whether the radicals have the same or different indices (root types).
Same Index Rule
a√b × c√d = (a × c)√(b × d)
Multiply coefficients outside and radicands inside separately
Different Index Rule
ᵐ√a × ⁿ√b = ᵏ√(aˢ × bᵗ)
Where k = LCM(m,n), s = k/m, t = k/n
Simplification Process
- 1.Find prime factorization of the radicand
- 2.Group factors by the index number
- 3.Extract complete groups as coefficients
- 4.Keep remaining factors under the radical
Example:
√72 = √(2³ × 3²) = √(2² × 2 × 3²)
= 2 × 3 × √2 = 6√2
Perfect Squares
Numbers with integer square roots:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
Perfect Cubes
Numbers with integer cube roots:
1, 8, 27, 64, 125, 216, 343, 512...
Common Factors
Useful for simplification:
2, 3, 5, 7, 11, 13, 17, 19, 23...