Question Detail
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}
1. \begin{aligned}
\text{if }6^m = 46656, \\\text{ What is the value of }6^{m-2}
\end{aligned}
- 7776
- 7782
- 1296
- 1290
Answer: Option C
Explanation:
\begin{aligned}
6^{m-2}\\ = \dfrac{6^m}{6^2}\\ = \dfrac{46656}{6^2}\\ = \dfrac{46656}{36} = 1296
\end{aligned}
2. \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}
3. \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}
4. Evaluate \begin{aligned} 256^{0.16} \times (256)^{0.09} \end{aligned}
- 2
- 4
- 8
- 16
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}
5. \begin{aligned} (1000)^7 \div (10)^{18} = ? \end{aligned}
- 10
- 100
- 1000
- 10000
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}