Solved Examples(Set 1) - Surds and Indices

Answer: Option B

Explanation:

(132)⁷ × (132)^x = (132)^11.5

=> 7 + x = 11.5

=> x = 11.5 - 7 = 4.5

Answer: Option A

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} $

Answer: Option D

Explanation:

7^{(x - y)} = 343 = 7³

=> x - y = 3 ---------------------------(Equation 1)

7^{(x + y)} = 16807 = 7⁵

=> x + y = 5 ---------------------------(Equation 2)

(Equation 1)+ (Equation 2) => 2x = 3 + 5 = 8

=> x = ⁸⁄₂ = 4

Answer: Option C

Explanation:

$\begin{align}&(0.04)^{-2.5} = \left(\dfrac{1}{.04}\right)^{2.5} = \left(\dfrac{100}{4}\right)^{2.5}

= \left(25\right)^{2.5} = \left(5^2\right)^{2.5}\\\\

&= \left(5^2\right)^{\left(\dfrac{5}{2}\right)}= 5^5 = 3125\end{align}$

Answer: Option A

Explanation:

(6)^6.5 × (36)^4.5 ÷ (216)^4.5

= (6)^6.5 × [(6)²]^4.5 ÷ [(6)³]^4.5

= (6)^6.5 × (6)⁹ ÷ (6)^13.5

= (6)^{(6.5 + 9 - 13.5)}

= (6)²