\[ \text { If } \frac{x^{12}+x^3}{x^6}=0, \text { find } x^{36}+\frac{1}{x^{36}} \]

Given:
– The equation \(\frac{x^{12} + x^3}{x^6} = 0\).

1. Simplify the equation by dividing each term by \(x^6\):
\[ \frac{x^{12}}{x^6} + \frac{x^3}{x^6} = 0 \]
\[ x^6 + \frac{1}{x^3} = 0 \]

2. Rearrange the equation to express \(x^6\) in terms of \(x^3\):
\[ x^6 = -\frac{1}{x^3} \]

3. Raise both sides of the equation to the 3rd power to eliminate the fraction:
\[ (x^6)^3 = \left(-\frac{1}{x^3}\right)^3 \]
\[ x^{18} = -1 \]

4. Further, raise both sides to the 2nd power to find \(x^{36}\):
\[ (x^{18})^2 = (-1)^2 \]
\[ x^{36} = 1 \]

Conclusion:
– The value of \(x^{36}\) is 1.
– Therefore, \(x^{36} + \frac{1}{x^{36}} = 1 + 1 = 2\).