Multiply Square Roots: √2 × √3 × √1 × √4 × √5 × √6 Step-by-Step

To solve this problem, we'll follow these steps:

• Step 1: Combine all the square root terms into a single square root using the product property of square roots.
• Step 2: Simplify the product inside the square root.
• Step 3: Simplify the square root expression obtained after combining.

Now, let's work through each step:
Step 1: We start by combining all the terms under one square root using the identity $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Thus:

$\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{1} \cdot \sqrt{4} \cdot \sqrt{5} \cdot \sqrt{6} = \sqrt{2 \cdot 3 \cdot 1 \cdot 4 \cdot 5 \cdot 6}$

Step 2: Calculate the product within the square root:
$2 \cdot 3 \cdot 1 \cdot 4 \cdot 5 \cdot 6 = 720$

Step 3: Now, simplify $\sqrt{720}$.
First, we find the prime factorization of 720: $720 = 2^4 \cdot 3^2 \cdot 5^1$.
Using the property that $\sqrt{a^2} = a$, we can write:

$\sqrt{720} = \sqrt{(2^2 \cdot 3)^2 \cdot 2 \cdot 5} = 2^2 \cdot 3 \cdot \sqrt{2 \cdot 5} = 4 \cdot 3 \cdot \sqrt{10}$

After simplification, the final answer is:

$4\sqrt{3}$.