GATE Civil 2017 Set 2 | GA | Question: 4

Question

What is the value of $x$ when $81\times\left (\dfrac{16}{25} \right )^{x+2}\div\left (\dfrac{3}{5} \right )^{2x+4}=144?$

• A. $1$
• B. $-1$
• C. $-2$
• D. $\text{Can not be determined}$

Comments

Easiest way is to substitute value,

If you try to smiplify it you may end up something like $(\frac{16}{9})$ $^{x+1}$ = 1 , only -1 stastify

Answer 1

$81\times\left (\frac{16}{25} \right )^{x+2}\div\left (\frac{3}{5} \right )^{2x+4}=144$

$\implies9^{2}\times\left (\frac{16}{25} \right )^{x+2}\div\left (\frac{3}{5} \right )^{2x+4}=144$

$\implies9^{2}\times\left (\frac{4}{5} \right )^{2(x+2)}\div\left (\frac{3}{5} \right )^{2(x+2)}=12^{2}$

$\implies\left [ 9\times\left (\frac{4}{5} \right )^{x+2}\div\left (\frac{3}{5} \right )^{x+2}\right ]^{2}=12^{2}$

$\implies\left [ 9\times\left (\frac{4}{5} \right )^{x+2}\div\left (\frac{3}{5} \right )^{x+2}\right ]=12$

$\implies \left [\left (\frac{4}{5} \right )^{x+2}\div\left (\frac{3}{5} \right )^{x+2}\right ]=\frac{4}{3}$

$\implies\frac{\left(\frac{4}{5}\right)^{x+2}}{\left(\frac{3}{5}\right)^{x+2}} = \frac{4}{3}$

$\implies\left(\frac{4}{3}\right)^{x+2}= \left(\frac{4}{3}\right)^{1}$

Compare both side and we get

     $x+2=1$

$\implies x=-1$

So, correct answer is option (B).

Answer 2

$\begin{align} 81\times\left( \frac{16}{25} \right)^{x+2}\div\left( \frac{3}{5} \right)^{2x+4}&=144\\ \Rightarrow  \left( \frac{16}{25} \right)^{x+2}\times\left( \frac{5}{3} \right)^{2(x+2)}&=\frac{144}{81}\\ \Rightarrow  \left( \frac{16}{25} \right)^{x+2}\times\left( \frac{25}{9} \right)^{x+2}&=\frac{16\times9}{9\times9}\\ \Rightarrow  \left( \frac{16}{25}\times\frac{25}{9} \right)^{x+2}&=\frac{16}{9}\\ \Rightarrow  \left( \frac{16}{9} \right)^{x+2}&=\frac{16}{9}\\ \Rightarrow x+2&=1\\ \therefore x&=-1\end{align}$

 

So the correct answer is B.

Answer 3

option b is right

Answer 4

Option B is right. 

Answer 5

$\text{By Prime Factorization,the given equation can be written as}$

$$ 3^4 \times \frac {2^{4x+8}}{5^{2x+4}} \times \frac {3^{-2x-4}}{5^{-2x-4}} = 3^2\times 2^4$$

$\text{comparing the powers of 2 on both sides}$

$$ \text{4x+8 = 4 }\implies x = -1$$

$$\text{(OR)}$$

$\text{comparing the powers of 3 on both sides}$

$$ \text{4-2x-4 = 2 }\implies x = -1$$