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. $\begin{aligned} (1000)^7 \div (10)^{18} = ? \end{aligned}$

10
100
1000
10000

Answer: Option C

Explanation:

$\begin{aligned} = \frac{(10^3)^7}{(10)^{18}} \end{aligned}$

$\begin{aligned} = \frac{(10)^{21}}{(10)^{18}} = 10^3 = 1000 \end{aligned}$

3. Find the value of
$\begin{aligned} (10)^{150} \div (10)^{146} \end{aligned}$

10
100
1000
10000

Answer: Option D

Explanation:

$\begin{aligned} = \frac{(10)^{150}}{(10)^{146}} = 10^4 = 10000 \end{aligned}$

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

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

Answer: Option D

Explanation:

$\begin{aligned} = (32)^{\frac{1}{3}}\\ = (2^5)^{\frac{1}{3}}\\ = 2^{\frac{5}{3}}\\ => x= \frac{5}{3} \end{aligned}$

5. 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,
$\begin{aligned} (11-1)^{2 + 1} = (10)^3 = 1000 \end{aligned}$