Answer :

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]