To rationalize the denominator of the expression \(\frac{15}{7\sqrt{3}}\), we multiply both the numerator and the denominator by \(\sqrt{3}\). This helps eliminate the square root from the denominator.
Here are the steps:
\[ \frac{15}{7\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{15\sqrt{3}}{7 \cdot 3} \]
Now simplify the denominator:
\[ 7 \cdot 3 = 21 \]
So, we have:
\[ \frac{15\sqrt{3}}{21} \]
Now, we can simplify the fraction \(\frac{15}{21}\):
The greatest common divisor (GCD) of 15 and 21 is 3.
\[ \frac{15}{21} = \frac{15 \div 3}{21 \div 3} = \frac{5}{7} \]
Thus, our expression simplifies to:
\[ \frac{5\sqrt{3}}{7} \]
So, the final answer is:
\[ \frac{5\sqrt{3}}{7} \]