Multiplying Radicals Calculator — Simplify Radical Products
Enter two radicands (values under the radical sign), their coefficients, and the root index. The calculator multiplies the radicals using the product rule ⁿ√a × ⁿ√b = ⁿ√(ab), then simplifies by extracting the largest perfect nth-power factor — and shows every step.
Root index
Exact simplified form shown in steps below
Multiply the coefficients
Apply the product rule — multiply the radicands
Extract the largest perfect factor from the radicand
Combine with the outer coefficient
Final simplified result
- 1
Multiply coefficients
2 × 3 = 6 - 2
Multiply radicands (product rule)
12 × 3 = 36 - 3
Extract largest perfect factor
√(36) = 6 · √(1) = 6 - 4
Combine with outer coefficient
6 × 6 = 36 - 5
Decimal result
36 × √(1) = 36
How does this calculator work?
Multiply radical expressions using c₁·ⁿ√a × c₂·ⁿ√b = (c₁·c₂)·ⁿ√(a·b), then simplify by pulling out the largest perfect nth-power factor. For example: 2√12 × 3√3 = 6√36 = 6·6 = 36. Works for square (√), cube (∛), and fourth roots (∜).
Formula
How this is calculated
The product rule for radicals states that ⁿ√a × ⁿ√b = ⁿ√(a·b) whenever both radicands are non-negative. This follows directly from the exponent product rule: a^(1/n) × b^(1/n) = (a·b)^(1/n). Coefficients in front of each radical multiply together independently: (3√5)(2√7) = 6√35.
After multiplying, the result is simplified by finding the largest perfect nth-power factor of the combined radicand. For a square root, this means the largest k² that divides the radicand — for example √72 = √(36·2) = 6√2, because 36 = 6² is the largest perfect square dividing 72. For a cube root, it is the largest k³ factor; for a fourth root, the largest k⁴ factor. The extracted value k is moved outside the radical, multiplying the existing coefficient.
This calculator handles only real-valued radicals: radicands must be non-negative (negative values under even-index roots produce complex numbers). Radicands should be integers for exact simplification; decimal radicands are evaluated numerically but the simplification step is skipped since fractional perfect powers are rarely meaningful in practice.
Frequently asked questions
The product rule ⁿ√a × ⁿ√b = ⁿ√(ab) requires the same index n on both radicals so the exponents 1/n combine cleanly: a^(1/n) × b^(1/n) = (ab)^(1/n). Radicals with different indices — say √5 × ∛5 — must first be rewritten with a common index (here 5^(3/6) × 5^(2/6) = 5^(5/6) = ⁶√(5⁵)) before combining.
For a square root √N, the calculator tries every integer k from ⌊√N⌋ downward, checking whether k² divides N evenly. The first k for which N mod k² = 0 is the largest — so √72 tries k=8 (64 doesn't divide 72), k=7 (49 doesn't), k=6 (36 divides 72) → 6 is the answer, giving 6√2. Cube and fourth roots use the same search with k³ and k⁴.
When the radicand is itself a perfect nth power — for example √144 = 12, or ∛27 = 3 — the inner radicand after extraction becomes 1, so the radical disappears. The result is simply the product of the two outer coefficients times k, shown as a plain integer in the simplified form.
Also known as
TG we-Calculate Editorial Team. (2026). Multiplying Radicals Calculator — Simplify Radical Products [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/multiplying-radicals-calculator
TG we-Calculate Editorial Team. "Multiplying Radicals Calculator — Simplify Radical Products." TG we-Calculate. 2026. https://we-calculate.com/calculator/multiplying-radicals-calculator.
TG we-Calculate Editorial Team, "Multiplying Radicals Calculator — Simplify Radical Products," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/multiplying-radicals-calculator
@misc{wecalculate_multiplying_radicals_calculator, title = {Multiplying Radicals Calculator — Simplify Radical Products}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/multiplying-radicals-calculator}}, year = {2026}, note = {TG we-Calculate} }
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