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

1. $1$
2. $-1$
3. $-2$
4. $\text{Can not be determined}$
Migrated from GO Civil 3 years ago by Arjun

1 comment

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

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

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

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

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

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

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

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

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

Compare both side and we get

$x+2=1$

$\implies x=-1$

So, correct answer is option (B).

\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.

option b is right

Option B is right.

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