\begin{aligned}
x = 3 + 2\sqrt{2}, \text{ then the value of }\\
(\sqrt{x} - \frac{1}{\sqrt{x}})
\end{aligned}

Answer: Option B

Explanation:

\begin{aligned}
= 256^{0.16+0.09} = 256^{0.25} = 256^{\frac{25}{100}}
\end{aligned}

\begin{aligned}
= 256^{\frac{1}{4}}= (4^4)^{\frac{1}{4}}
\end{aligned}

\begin{aligned}
=(4)^{4 \times \frac{1}{4}} = 4
\end{aligned}

Answer: Option A

Explanation:

\begin{aligned}
= \frac{1}{\left( 1 + \frac{a^n}{a^m} \right)} +
\frac{1}{\left( 1 + \frac{a^m}{a^n} \right)} \\
= \frac{a^m}{(a^m+a^n)} + \frac{a^n}{(a^m+a^n)} \\
= \frac{(a^m+a^n)}{(a^m+a^n)} = 1
\end{aligned}

Answer: Option B

Explanation:

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

Answer: Option D

Explanation:

We know that \begin{aligned} (11)^2 = 121
\end{aligned}
So, putting values in said equation we get,
\begin{aligned} (11-1)^{2 + 1} = (10)^3 = 1000 \end{aligned}

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}