Answer: To simplify the given expression, let's start by combining the like terms:
3√2 √6 - √3 4√3 √6 - √2 + 2√3 √6 + 2
Now, let's work on each pair of like terms separately:
3√2 √6 - √2:
To simplify this, we can factor out √2 from both terms:
3√2 √6 - √2 = √2 (3√6 - 1)
√3 4√3 √6:
Here, we can factor out √3 from both terms:
√3 4√3 √6 = -√3 (4√6 + 1)
2√3 √6 + 2:
In this case, we can also factor out √3 from both terms:
2√3 √6 + 2 = √3 (2√6 + 2)
Now, let's put all the simplified terms together:
√2 (3√6 - 1) - √3 (4√6 + 1) + √3 (2√6 + 2)
Next, we can distribute the √2 and √3:
√2 * 3√6 - √2 * 1 - √3 * 4√6 - √3 * 1 + √3 * 2√6 + √3 * 2
Now, we can simplify the products of square roots:
3√(2 * 6) - √2 - 4√(3 * 6) - √3 + 2√(3 * 6) + 2√3
3√12 - √2 - 4√18 - √3 + 2√18 + 2√3
Now, let's simplify the square roots:
3√(2 * 2 * 3) - √2 - 4√(2 * 3 * 3) - √3 + 2√(2 * 3 * 3) + 2√3
3√2^2 √3 - √2 - 4√2^2 √3 - √3 + 2√2^2 √3 + 2√3
3 * 2√3 - √2 - 4 * 2√3 - √3 + 2 * 2√3 + 2√3
6√3 - √2 - 8√3 - √3 + 4√3 + 2√3
Now, let's combine the like terms again:
(6√3 - 8√3 + 4√3) - (√2 + √3) + 2√3
2√3 - (√2 + √3) + 2√3
Finally, let's combine the remaining like terms:
(2√3 + 2√3) - √2 - √3
4√3 - √2 - √3
So, the simplified expression is 4√3 - √2 - √3.