Multiplying Radicals: The Complete Guide to Radical Product Simplification
Master multiplying radicals with the product rule ⁿ√a × ⁿ√b = ⁿ√(ab). Learn formulas, worked examples, and simplification techniques.
Try the free calculator
Put these formulas into practice with our instant, step-by-step Multiplying Radicals Calculator.
Multiplying radicals combines two root expressions into one using the product rule: ⁿ√a × ⁿ√b = ⁿ√(a×b). This fundamental technique simplifies radical expressions and is essential for algebra, geometry, and scientific calculations involving square roots, cube roots, and higher-order radicals.
Key Takeaway
When multiplying radicals with the same index, multiply the radicands: ⁿ√a × ⁿ√b = ⁿ√(ab). Always simplify the result by factoring out perfect powers.
1. The Product Rule for Radicals
This rule works because radicals can be expressed as fractional exponents: ⁿ√a = a^(1/n). When multiplying, a^(1/n) × b^(1/n) = (ab)^(1/n) = ⁿ√(ab).
2. Worked Examples
Example 1: √3 × √12
- Apply product rule: √3 × √12 = √(3 × 12) = √36
- Simplify: √36 = 6
- Verify: 1.732 × 3.464 ≈ 6 —
Example 2: With coefficients
- 2√3 × 3√5 = (2 × 3) × (√3 × √5)
- Coefficients: 2 × 3 = 6
- Radicals: √3 × √5 = √15
- Result: 6√15
3. Simplification Techniques
After multiplying, always check if the radicand has perfect power factors. For example, √50 = √(25 × 2) = 5√2.
| Expression | Simplified |
|---|---|
| √12 × √3 | √36 = 6 |
| √8 × √2 | √16 = 4 |
| ³√5 × ³√25 | ³√125 = 5 |
4. Different Indices
When radicals have different indices (e.g., √a × ³√b), convert to fractional exponents first: a^(1/2) × b^(1/3). Then find a common denominator or compute numerically.
5. Frequently Asked Questions
What is √a × √a?
√a × √a = a. Any square root multiplied by itself equals the radicand.
Can I multiply radicals with variables?
Yes. √(2x) × √(3x) = √(6x²) = x√6 (assuming x > 0).
© 2026 Calculator Archive. Free online calculators for math, finance, health, and more.