To simplify the expression \( \frac{15}{7\sqrt{3}} \) and rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \):
\[ \frac{15}{7\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{15\sqrt{3}}{7 \cdot 3} \]
Now, simplify the denominator:
\[ 7 \cdot 3 = 21 \]
So the expression now is:
\[ \frac{15\sqrt{3}}{21} \]
Next, simplify \( \frac{15}{21} \):
Both 15 and 21 can be divided by 3:
\[ \frac{15 \div 3}{21 \div 3} = \frac{5}{7} \]
Thus, we can express the simplified expression as:
\[ \frac{5\sqrt{3}}{7} \]
So the final, simplified result is:
\[ \frac{5\sqrt{3}}{7} \]