Certainly! Let's simplify the given surd expression step by step:
[tex]\[
\left( \frac{\sqrt{3}}{2 \sqrt{3}} \right) \times \left( \frac{2 \sqrt{2} + 2 \sqrt{2}}{1 \sqrt{2} + 2 \sqrt{3}} \right)
\][/tex]
### First Part: [tex]\(\frac{\sqrt{3}}{2 \sqrt{3}}\)[/tex]
1. Simplify the numerator and the denominator:
- Numerator: [tex]\(\sqrt{3}\)[/tex]
- Denominator: [tex]\(2\sqrt{3}\)[/tex]
2. Divide the numerator by the denominator:
[tex]\[
\frac{\sqrt{3}}{2\sqrt{3}} = \frac{1}{2}
\][/tex]
So, the first part simplifies to [tex]\(\frac{1}{2}\)[/tex].
### Second Part: [tex]\(\frac{2 \sqrt{2} + 2 \sqrt{2}}{1 \sqrt{2} + 2 \sqrt{3}}\)[/tex]
1. Simplify the numerator:
[tex]\[
2 \sqrt{2} + 2 \sqrt{2} = 4 \sqrt{2}
\][/tex]
2. The denominator remains:
[tex]\[
\sqrt{2} + 2 \sqrt{3}
\][/tex]
So, the second part simplifies to [tex]\(\frac{4 \sqrt{2}}{\sqrt{2} + 2 \sqrt{3}}\)[/tex].
### Combine the Simplified Parts
1. Multiply the simplified first part by the numerator of the second part:
[tex]\[
\left( \frac{1}{2} \right) \times 4 \sqrt{2} = 2 \sqrt{2}
\][/tex]
2. The final expression is:
[tex]\[
\frac{2\sqrt{2}}{\sqrt{2} + 2\sqrt{3}}
\][/tex]
### Final Result
The simplified result of the given surd expression is approximately:
[tex]\[
0.5797958971132713
\][/tex]
Additionally, in the intermediate steps we have:
- The simplified value of [tex]\(\frac{\sqrt{3}}{2 \sqrt{3}}\)[/tex] is [tex]\(0.5\)[/tex].
- The value of [tex]\(4 \sqrt{2}\)[/tex] is approximately [tex]\(5.656854249492381\)[/tex].
Thus, the step-by-step simplification of the given surd expression is now complete.
