To rationalize the denominator of the expression [tex]\(\frac{15+\sqrt{3}}{10 \sqrt{3}}\)[/tex], we need to eliminate the square root in the denominator.
First, we multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\frac{15+\sqrt{3}}{10\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{(15+\sqrt{3})\sqrt{3}}{10(\sqrt{3}\sqrt{3})}
\][/tex]
Simplify the denominator:
[tex]\[
10(\sqrt{3}\sqrt{3}) = 10 \cdot 3 = 30
\][/tex]
Now, distribute [tex]\(\sqrt{3}\)[/tex] in the numerator:
[tex]\[
(15 + \sqrt{3})\sqrt{3} = 15\sqrt{3} + \sqrt{3}\sqrt{3} = 15\sqrt{3} + 3
\][/tex]
So the expression becomes:
[tex]\[
\frac{15\sqrt{3} + 3}{30}
\][/tex]
Next, split the fraction into two separate fractions:
[tex]\[
\frac{15\sqrt{3}}{30} + \frac{3}{30}
\][/tex]
Simplify each fraction:
[tex]\[
\frac{15\sqrt{3}}{30} = \frac{15}{30} \sqrt{3} = \frac{1}{2} \sqrt{3}
\][/tex]
[tex]\[
\frac{3}{30} = \frac{1}{10}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{1}{2} \sqrt{3} + \frac{1}{10}
\][/tex]
Combining these results, the rationalized and simplified form of the given expression [tex]\(\frac{15+\sqrt{3}}{10\sqrt{3}}\)[/tex] is:
[tex]\[
\frac{\sqrt{3}}{2} + \frac{1}{10}
\][/tex]
