To simplify the expression [tex]\( \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{2}}{\sqrt{3}} \)[/tex] to its simplest radical form, follow these detailed steps.
### Step 1: Simplify Each Fraction
1. Simplify [tex]\(\frac{6}{\sqrt{3}}\)[/tex]:
- First, rationalize the fraction by multiplying both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}
\][/tex]
2. Simplify [tex]\(\frac{\sqrt{2}}{\sqrt{3}}\)[/tex]:
- Rationalize this fraction by also multiplying both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{2} \cdot \sqrt{3}}{3} = \frac{\sqrt{6}}{3}
\][/tex]
### Step 2: Combine the Simplified Parts
- Now multiply the simplified results together:
[tex]\[
2\sqrt{3} \cdot \frac{\sqrt{6}}{3}
\][/tex]
### Step 3: Perform Multiplication and Simplify Further
- Combine the terms into a single fraction:
[tex]\[
\frac{2\sqrt{3} \cdot \sqrt{6}}{3}
\][/tex]
- Next, simplify the numerator:
[tex]\[
2 \cdot \sqrt{3 \cdot 6} = 2 \cdot \sqrt{18}
\][/tex]
- This can be rewritten by simplifying [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\][/tex]
- Thus:
[tex]\[
2 \cdot 3\sqrt{2} = 6\sqrt{2}
\][/tex]
- Now divide by the 3 in the denominator:
[tex]\[
\frac{6\sqrt{2}}{3} = 2\sqrt{2}
\][/tex]
### Final Simplified Form
Thus, the simplified form of the expression [tex]\(\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{2}}{\sqrt{3}}\)[/tex] is:
[tex]\[
\boxed{2\sqrt{2}}
\][/tex]
