Surds and Indices Problems and Solutions

• 1. \begin{aligned} \sqrt{8}^\frac{1}{3} \end{aligned}

  1. 2
  2. 4
  3. \begin{aligned} \sqrt{2} \end{aligned}
  4. 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}

• 2. \begin{aligned}
  \text{if }6^m = 46656, \\\text{ What is the value of }6^{m-2}
  \end{aligned}

  1. 7776
  2. 7782
  3. 1296
  4. 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}

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

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

  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 :

  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. \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 :

  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 :)

• 6. 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. 1
  2. 10
  3. 100
  4. 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}

• 7. \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 :

  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}