Certainly! To simplify the expression \(\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{2}}{\sqrt{3}}\), let's go through the steps in detail:
1. Combine the fractions:
[tex]\[
\frac{6 \cdot \sqrt{2}}{\sqrt{3} \cdot \sqrt{3}}
\][/tex]
2. Multiply the terms in the denominator:
[tex]\[
\sqrt{3} \cdot \sqrt{3} = 3
\][/tex]
3. So the expression now is:
[tex]\[
\frac{6 \cdot \sqrt{2}}{3}
\][/tex]
4. Simplify the fraction \(\frac{6}{3}\):
[tex]\[
\frac{6 \cdot \sqrt{2}}{3} = 2 \cdot \sqrt{2}
\][/tex]
5. Calculate the value of \( \sqrt{2} \):
[tex]\[
\sqrt{2} \approx 1.4142135623730951
\][/tex]
6. Calculate the final result:
[tex]\[
2 \times 1.4142135623730951 \approx 2.8284271247461903
\][/tex]
So the simplified form of the expression \(\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{2}}{\sqrt{3}}\) is \(2\sqrt{2}\).
The numerical result is therefore:
[tex]\[
2.8284271247461903
\][/tex]
Thus, [tex]\(\boxed{2\sqrt{2}}\)[/tex] is the simplest radical form of the given expression.
