136. \( \left(\frac{x^{-2} y^{3} z}{x^{4} y z^{-3}}\right)^{3}=\frac{z^{12}}{x^{18} y^{6}} \)

To solve the equation \( \left(\frac{x^{-2} y^{3} z}{x^{4} y z^{-3}}\right)^{3}=\frac{z^{12}}{x^{18} y^{6}} \), let's simplify the left side first. 1. **Simplifying the fraction:** \[ \frac{x^{-2} y^{3} z}{x^{4} y z^{-3}} = \frac{x^{-2}}{x^{4}} \cdot \frac{y^{3}}{y} \cdot \frac{z}{z^{-3}} = x^{-2-4} \cdot y^{3-1} \cdot z^{1-(-3)} \] \[ = x^{-6} \cdot y^{2} \cdot z^{4} = \frac{y^{2} z^{4}}{x^{6}} \] 2. **Raising to the power of 3:** \[ \left(\frac{y^{2} z^{4}}{x^{6}}\right)^{3} = \frac{y^{2 \cdot 3} z^{4 \cdot 3}}{x^{6 \cdot 3}} = \frac{y^{6} z^{12}}{x^{18}} \] 3. **Now we equate both sides:** \[ \frac{y^{6} z^{12}}{x^{18}} = \frac{z^{12}}{x^{18} y^{6}} \] 4. **Since the denominators \(x^{18}\) are the same, we can equate the numerators:** \[ y^{6} z^{12} = z^{12} y^{-6} \] 5. **Dividing both sides by \(z^{12}\) (assuming \(z \neq 0\)) results in:** \[ y^{6} = y^{-6} \] 6. **Multiplying through by \(y^{6}\) gives:** \[ y^{12} = 1 \] 7. **Thus, \(y^{6} = 1\), implying \(y = 1\) or \(y = -1\).** Now, summarizing, The solution gives us \(y = 1\) or \(y = -1\), provided that other variables satisfy their algebraic conditions within a given context. Therefore, the solution to the equation is \(y = \pm 1\).