Surds and Indices Questions Answers

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

  1. 1.5
  2. 4.5
  3. 7.5
  4. 9.5

  Answer And Explanation

  Answer: Option B

  Explanation:

  $$\begin{align}&\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}\\\\ &\Rightarrow \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{a}{b}\right)^{-(x-7)}\\\\ &\Rightarrow x - 2 = -(x - 7)\\\\ &\Rightarrow x - 2 = -x + 7\\\\ &\Rightarrow x-2 = -x + 7\\\\ &\Rightarrow 2x = 9\\\\ &\Rightarrow x = \dfrac{9}{2} = 4.5 \end{align}$$

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

  1. $$\begin{aligned} \sqrt{13} \end{aligned}$$
  2. $$\begin{aligned} \sqrt{14} \end{aligned}$$
  3. $$\begin{aligned} \sqrt{15} \end{aligned}$$
  4. $$\begin{aligned} \sqrt{16} \end{aligned}$$

  Answer And Explanation

  Answer: Option B

  Explanation:

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

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

  1. 25
  2. 28
  3. 36
  4. 44

  Answer And Explanation

  Answer: Option C

  Explanation:

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

  1. $$\begin{aligned} \frac{5}{2} \end{aligned}$$
  2. $$\begin{aligned} \frac{2}{5} \end{aligned}$$
  3. $$\begin{aligned} \frac{3}{5} \end{aligned}$$
  4. $$\begin{aligned} \frac{5}{3} \end{aligned}$$

  Answer And Explanation

  Answer: Option D

  Explanation:

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

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

  1. 1
  2. 2
  3. 3
  4. 4

  Answer And Explanation

  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 :)

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• 13. $$\begin{aligned} \frac{1}{1+a^{(n-m)}} + \frac{1}{1+a^{(m-n)}} = ? \end{aligned}$$

  1. 1
  2. 2
  3. 3
  4. 4

  Answer And Explanation

  Answer: Option A

  Explanation:

  $$\begin{aligned} = \frac{1}{\left( 1 + \frac{a^n}{a^m} \right)} + \frac{1}{\left( 1 + \frac{a^m}{a^n} \right)} \\ = \frac{a^m}{(a^m+a^n)} + \frac{a^n}{(a^m+a^n)} \\ = \frac{(a^m+a^n)}{(a^m+a^n)} = 1 \end{aligned}$$

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• 14. $$\begin{aligned} \text{If } 3^{x-y} = 27 \text{ and } 3^{x+y} = 243, \\ \text{ then find the value of x } \end{aligned}$$

  1. 1
  2. 2
  3. 3
  4. 4

  Answer And Explanation

  Answer: Option D

  Explanation:

  $$\begin{aligned}3^{x-y} = 27 = 3^3 <=> x-y = 3 \text{... (i)}\\ 3^{x+y} = 243 = 3^5 <=> x+y = 5 \text{... (ii)} \\ \text{ adding (i) and (ii)} => 2x = 8 \\ => x = 4 \end{aligned}$$

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