The problem is to simplify the expression: $\frac{\frac{\sqrt{3}}{3} \times 1 + \frac{\sqrt{3}}{3} \times (-1) + \frac{\sqrt{3}}{3} \times 0}{\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} \times \sqrt{1^2 + (-1)^2 + 0^2}}$

1. Problem Description

The problem is to simplify the expression:

$\frac{\frac{\sqrt{3}}{3} \times 1 + \frac{\sqrt{3}}{3} \times (-1) + \frac{\sqrt{3}}{3} \times 0}{\sqrt{(\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2 + (\frac{\sqrt{3}}{3})^2} \times \sqrt{1^2 + (-1)^2 + 0^2}}$

2. Solution Steps

First, simplify the numerator:

$\frac{\sqrt{3}}{3} \times 1 + \frac{\sqrt{3}}{3} \times (-1) + \frac{\sqrt{3}}{3} \times 0 = \frac{\sqrt{3}}{3} - \frac{\sqrt{3}}{3} + 0 = 0$

Next, simplify the denominator:

The first radical is

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

The second radical is

$\sqrt{1^2 + (-1)^2 + 0^2} = \sqrt{1 + 1 + 0} = \sqrt{2}$

Therefore, the denominator is

$1 \times \sqrt{2} = \sqrt{2}$

The entire expression becomes:

$\frac{0}{\sqrt{2}} = 0$

3. Final Answer

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