Question Detail

\begin{aligned}
\text{If }2x = \sqrt[3]{32}, \text{ then x is equal to}
\end{aligned}

\begin{aligned} \frac{5}{2} \end{aligned}
\begin{aligned} \frac{2}{5} \end{aligned}
\begin{aligned} \frac{3}{5} \end{aligned}
\begin{aligned} \frac{5}{3} \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}

1. Find the value of,
\begin{aligned}
\frac{1}{216^{-\frac{2}{3}}}+\frac{1}{256^{-\frac{3}{4}}}+\frac{1}{32^{-\frac{1}{5}}}
\end{aligned}

Answer: Option C

2. \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}

3. \begin{aligned}
\text{If }x = \left(8 + 3\sqrt{7}\right),\text{ what is the value of }\\\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)?
\end{aligned}

\begin{aligned} \sqrt{13} \end{aligned}
\begin{aligned} \sqrt{14} \end{aligned}
\begin{aligned} \sqrt{15} \end{aligned}
\begin{aligned} \sqrt{16} \end{aligned}

Answer: Option B

Explanation:

\begin{align}&\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2\\\\
&= x - 2 + \dfrac{1}{x}\\\\
&= x + \dfrac{1}{x} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{1}{\left(8 + 3\sqrt{7}\right)} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{\left(8 + 3\sqrt{7}\right)\left(8 - 3\sqrt{7}\right)} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{8^2 - \left(3\sqrt{7}\right)^2} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{64 - 63} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{1} - 2 \\\\
&= 8 + 3\sqrt{7} + 8 - 3\sqrt{7} - 2 \\\\
&= 14 \\\\
&\text{as }\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2 = 14\\\\
&\text{so ,}\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right) = \sqrt{14}\end{align}

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} \text{If } 5^{(a + b)} = 5 \times 25 \times 125 ,\\ \text{what is }(a + b)^2

\end{aligned}

Answer: Option C

Question Detail

\begin{aligned} \text{If }2x = \sqrt[3]{32}, \text{ then x is equal to} \end{aligned}

1. Find the value of, \begin{aligned} \frac{1}{216^{-\frac{2}{3}}}+\frac{1}{256^{-\frac{3}{4}}}+\frac{1}{32^{-\frac{1}{5}}} \end{aligned}

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

3. \begin{aligned} \text{If }x = \left(8 + 3\sqrt{7}\right),\text{ what is the value of }\\\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)? \end{aligned}

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

5. \begin{aligned} \text{If } 5^{(a + b)} = 5 \times 25 \times 125 ,\\ \text{what is }(a + b)^2 \end{aligned}

\begin{aligned}
\text{If }2x = \sqrt[3]{32}, \text{ then x is equal to}
\end{aligned}

1. Find the value of,
\begin{aligned}
\frac{1}{216^{-\frac{2}{3}}}+\frac{1}{256^{-\frac{3}{4}}}+\frac{1}{32^{-\frac{1}{5}}}
\end{aligned}

3. \begin{aligned}
\text{If }x = \left(8 + 3\sqrt{7}\right),\text{ what is the value of }\\\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)?
\end{aligned}

5. \begin{aligned} \text{If } 5^{(a + b)} = 5 \times 25 \times 125 ,\\ \text{what is }(a + b)^2

\end{aligned}