Question Detail

\begin{aligned}
\frac{1}{1+a^{(n-m)}} + \frac{1}{1+a^{(m-n)}} = ?
\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}

1. \begin{align}
\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}.\\\text{ What is the value of x ?}
\end{align}

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}

2. Evaluate \begin{aligned} 256^{0.16} \times (256)^{0.09} \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}

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

Answer: Option B

Explanation:

Clue:

\begin{aligned}
(\sqrt{x} - \frac{1}{\sqrt{x}})^2 = x + \frac{1}{x} - 2 \
\end{aligned}
Now put the value of x to calculate the answer :)

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

1012
1024
1048
525

Answer: Option B

Explanation:

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

5. \begin{aligned} \sqrt{8}^\frac{1}{3} \end{aligned}

2
4
\begin{aligned} \sqrt{2} \end{aligned}
8

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}