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:

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

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:

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

Now, the denominator becomes:

$1 \times 3 = 3$

Therefore, the whole expression becomes:

$\frac{\frac{\sqrt{3}}{3}}{3} = \frac{\sqrt{3}}{3} \times \frac{1}{3} = \frac{\sqrt{3}}{9}$

3. Final Answer

$\frac{\sqrt{3}}{9}$