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]
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]