To simplify the given expression:
[tex]\[ \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{2}}{\sqrt{3}} \][/tex]
First, let's rewrite the expression by combining the fractions:
[tex]\[ \frac{6 \cdot \sqrt{2}}{\sqrt{3} \cdot \sqrt{3}} \][/tex]
Next, let's simplify the denominator:
[tex]\[ \sqrt{3} \cdot \sqrt{3} = 3 \][/tex]
So the expression now is:
[tex]\[ \frac{6 \cdot \sqrt{2}}{3} \][/tex]
Let's simplify the fraction:
[tex]\[ \frac{6}{3} = 2 \][/tex]
Thus, the expression now is:
[tex]\[ 2 \cdot \sqrt{2} \][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{2}}{\sqrt{3}}\)[/tex] is:
[tex]\[ 2\sqrt{2} \][/tex]
