Simplify; $\frac{3\sqrt{5} \times 4\sqrt{6}}{2 \sqrt{3} \times 3\sqrt{2}}$

Step 1: Analyze the question We need to simplify a fraction made of products of surds. The approach is to multiply the whole numbers together, multiply the surds together, then simplify.

Step 2: Concept

• Rule: √a × √b = √(ab)
• Also: √a / √b = √(a/b)

Step 3: Solution

Numerator: 3√5 × 4√6 = (3 × 4) × √(5 × 6) = 12√30

Denominator: 2√3 × 3√2 = (2 × 3) × √(3 × 2) = 6√6

Now divide:

12√30 / 6√6 = (12/6) × √(30/6) = 2 × √5 = 2√5

Step 4: Answer The result is 2√5, which is option D.

Tip: Before multiplying out, always separate the whole numbers from the surds. Divide the whole numbers (12 ÷ 6 = 2), then combine the surds under one root (√30 ÷ √6 = √5). Cancelling early keeps numbers small and reduces mistakes under exam pressure.