1. **Problem statement:** Given the exponential expressions $P = a^x$, $Q = a^y$, and $R = a^z$, prove the following: (ক) If $a^x = b$, $b^y = c$, and $c^z = a$, then show $xyz = 1$. 2. **Formula and rules:** Recall the properties of exponents: - $ (a^m)^n = a^{mn} $ - If $a^m = a^n$, then $m = n$ (assuming $a > 0$ and $a \neq 1$). 3. **Step-by-step solution for (ক):** - From $a^x = b$, raise both sides to the power $y$: $$b^y = (a^x)^y = a^{xy}$$ - Given $b^y = c$, so $c = a^{xy}$. - Now, $c^z = a$ implies: $$ (a^{xy})^z = a \Rightarrow a^{xyz} = a^1 $$ - Since bases are equal and $a > 0, a \neq 1$, exponents must be equal: $$ xyz = 1 $$ 4. **Final answer for (ক):** $xyz = 1$. **Note:** Only the first problem is solved as per instructions.