Question

Solve for $x$ : ${\left(\frac{9}{4}\right)}^x\times {\left(\frac{8}{27}\right)}^{x-1}=\frac{2}{3}$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

${\left(\frac{9}{4}\right)}^x\times {\left(\frac{8}{27}\right)}^{x-1}=\frac{2}{3}$
$\Rightarrow {\left(\frac{3}{2}\right)}^{2x}\times {\left(\frac{2}{3}\right)}^{3(x-1)}=\frac{2}{3}$
$\Rightarrow {\left(\frac{2}{3}\right)}^{-2x}\times {\left(\frac{2}{3}\right)}^{3x-3}=\frac{2}{3}$
$\Rightarrow -2x+3x-3=1$

$\Rightarrow x-3=1$

$\Rightarrow x=4$