The problem is to simplify the expression: $\frac{\frac{\sqrt{3}}{3} \times (-2) + \frac{\sqrt{3}}{3} \times (-2) + \frac{\sqrt{3}}{3} \times 1}{\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} \times \sqrt{(-2)^2 + (-2)^2 + 1^2}}$

1. Problem Description

The problem is to simplify the expression:

$\frac{\frac{\sqrt{3}}{3} \times (-2) + \frac{\sqrt{3}}{3} \times (-2) + \frac{\sqrt{3}}{3} \times 1}{\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} \times \sqrt{(-2)^2 + (-2)^2 + 1^2}}$

2. Solution Steps

First, simplify the numerator:

$\frac{\sqrt{3}}{3} \times (-2) + \frac{\sqrt{3}}{3} \times (-2) + \frac{\sqrt{3}}{3} \times 1 = \frac{\sqrt{3}}{3}(-2 - 2 + 1) = \frac{\sqrt{3}}{3}(-3) = -\sqrt{3}$

Next, simplify the first square root in the denominator:

$\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} = \sqrt{\frac{3}{9} + \frac{3}{9} + \frac{3}{9}} = \sqrt{\frac{9}{9}} = \sqrt{1} = 1$

Then, simplify the second square root in the denominator:

$\sqrt{(-2)^2 + (-2)^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$

Therefore, the denominator is $1 \times 3 = 3$.

The expression becomes:

$\frac{-\sqrt{3}}{3}$

3. Final Answer

$-\frac{\sqrt{3}}{3}$