1. **Problem:** Given $a = b^x$, $b = c^y$, and $c = a^z$, prove that $xyz = 1$. 2. **Step 1:** Express $a$ in terms of $a$ using the given equations. From $a = b^x$ and $b = c^y$, substitute $b$: $$a = (c^y)^x = c^{xy}$$ From $c = a^z$, substitute $c$: $$a = (a^z)^{xy} = a^{xyz}$$ 3. **Step 2:** Since $a = a^{xyz}$ and $a \neq 0,1$, equate the exponents: $$1 = xyz$$ 4. **Conclusion:** Hence, we have proved that $$xyz = 1$$ This completes the proof.