Nice question!

Here's another approach:

Let's first deal with the denominator: 1/√2 + 1/√3 = √3/√6 + √2/√6
= (√3 + √2)/√6

So....\(\frac{\sqrt{6}}{(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}})}=\) √6/[(√3 + √2)/√6]

= (√6)(√6)/(√3 + √2)

= 6/(√3 + √2)

Now we can use the fact that √3 ≈ 1.7 and √2 ≈ 1.4 to get....
≈ 6/(1.7 + 1.4)
≈ 6/(3.1)

Since 6/3 = 2, we know that 6/(3.1) = a number a bit smaller than 2

Now check the answer choices to see which one evaluates to be a number a bit smaller than 2

A. \(\frac{6}{5}\) ELIMINATE

B. \(\frac{5}{6}\) ELIMINATE

C. 5 ELIMINATE

D. \(2\sqrt{3} +3\sqrt{2}\) ≈ 2(1.7) + 3(1.4) ≈ a number that's bigger than 6 ELIMINATE

E. \(6\sqrt{3}-6\sqrt{2}\) = 6(√3 - √2) ≈ 6(1.7 - 1.4) ≈ 6(0.3) ≈ 1.8 PERFECT!

Answer: E