Surds and Indices Problems and Solutions

• 8. \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 :

  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}

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

  1. 10
  2. 100
  3. 1000
  4. 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}

• 10. \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 :

  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}

• 11. Evaluate \begin{aligned} 256^{0.16} \times (256)^{0.09} \end{aligned}

  1. 2
  2. 4
  3. 8
  4. 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}

• 12. \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 :

  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}

• 13. \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 :

  Option C

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

  1. 1012
  2. 1024
  3. 1048
  4. 525

  Answer :

  Option B

  Explanation:

  \begin{aligned}
  = (256)^{\frac{5}{4}} = (4^4)^{\frac{5}{4}} = 4^5 = 1024
  \end{aligned}