Value of \begin{aligned} (256)^{\frac{5}{4}} \end{aligned}

Answer: Option C

Explanation:

\begin{aligned}
= ((8)^\frac{1}{2})^\frac{1}{3} = 8^{(\frac{1}{2} \times \frac{1}{3})}
\end{aligned}

\begin{aligned}
= (8)^{\frac{1}{6}}
\end{aligned}

\begin{aligned}
= (2)^{3(\frac{1}{6})}
\end{aligned}

\begin{aligned}
= (2)^{\frac{1}{2}}
\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{align}&\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}\\\\
&\Rightarrow \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{a}{b}\right)^{-(x-7)}\\\\
&\Rightarrow x - 2 = -(x - 7)\\\\
&\Rightarrow x - 2 = -x + 7\\\\
&\Rightarrow x-2 = -x + 7\\\\
&\Rightarrow 2x = 9\\\\
&\Rightarrow x = \dfrac{9}{2} = 4.5
\end{align}

Answer: Option D

Explanation:

Lets assume,
\begin{aligned}
\left(25 \right)^{7.5} \times \left(5 \right)^{2.5} \div \left(125 \right)^{1.5} = 5^x \\
\text{then, } \frac{ \left( 5^2 \right)^{7.5} \times \left(5 \right)^{2.5} }{\left(5^3 \right)^{1.5} } = 5^x \\
=> \text{then, } \frac{ \left( 5^{15} \right) \times \left(5^{2.5} \right) }{\left(5^{4.5} \right) } = 5^x \\

=> 5^x = 5^{15 + 2.5 - 4.5} \\
=> 5^x = 5^{13} \\
\text{Hence, } x = 13
\end{aligned}

Answer: Option C

Explanation:

\begin{aligned}
6^{m-2}\\ = \dfrac{6^m}{6^2}\\ = \dfrac{46656}{6^2}\\ = \dfrac{46656}{36} = 1296
\end{aligned}