Question Detail
\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}
1. \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}
2. \begin{aligned}
x = 3 + 2\sqrt{2}, \text{ then the value of }\\
(\sqrt{x} - \frac{1}{\sqrt{x}})
\end{aligned}
- 1
- 2
- 3
- 4
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 :)
3. If m and n are whole numbers such that
\begin{aligned} m^n=121 \end{aligned}
, the value of \begin{aligned} (m-1)^{n + 1} \end{aligned} is
- 1
- 10
- 100
- 1000
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}
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}
\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}