\begin{aligned}
\text{If } 3^{x-y} = 27 \text{ and } 3^{x+y} = 243, \\
\text{ then find the value of x }
\end{aligned}

Answer: Option D

Explanation:

\begin{aligned}
= (32)^{\frac{1}{3}}\\
= (2^5)^{\frac{1}{3}}\\
= 2^{\frac{5}{3}}\\
=> x= \frac{5}{3}
\end{aligned}

Answer: Option C

Explanation:

\begin{aligned}
= \frac{(10^3)^7}{(10)^{18}}
\end{aligned}

\begin{aligned}
= \frac{(10)^{21}}{(10)^{18}} = 10^3 = 1000
\end{aligned}

Answer: Option D

Explanation:

\begin{aligned}
= \frac{(10)^{150}}{(10)^{146}} = 10^4 = 10000
\end{aligned}

Answer: Option D

Explanation:

\begin{aligned}3^{x-y} = 27 = 3^3 <=> x-y = 3 \text{... (i)}\\
3^{x+y} = 243 = 3^5 <=> x+y = 5 \text{... (ii)} \\

\text{ adding (i) and (ii)}
=> 2x = 8 \\
=> x = 4
\end{aligned}

Answer: Option B

Explanation:

\begin{aligned}
= (256)^{\frac{5}{4}} = (4^4)^{\frac{5}{4}} = 4^5 = 1024
\end{aligned}