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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}

  • 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}

  • 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

  • 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}

  • 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 :)

  • 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}

  • 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}