To solve the equation \( x^2 = \frac{36}{169} \), we can take the square root of both sides.
First, we rewrite the equation:
\[ x^2 = \frac{36}{169} \]
Taking the square root of both sides, we have:
\[ x = \pm \sqrt{\frac{36}{169}} \]
This simplifies to:
\[ x = \pm \frac{\sqrt{36}}{\sqrt{169}} = \pm \frac{6}{13} \]
Thus, the solution set is:
\[ \left{ \frac{6}{13}, -\frac{6}{13} \right} \]