Evaluate : $3^{2 x - 3} - 9^{x - 1} + 27^{2 x / 3} = 675$

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Solution

$3^{2 x - 3} - 9^{x - 1} + 27^{2 x / 3} = 675$
$\Rightarrow 3^{2 x - 3} - 3^{2 (x - 1)} + 3^{3 \times \frac{2 x}{3}} = 675$
$\Rightarrow \frac{3^{2 x}}{3^{3}} - \frac{3^{2 x}}{3^{2}} + 3^{2 x} = 675$
$\Rightarrow 3^{2 x} [\frac{1}{27} + 1 - \frac{1}{9}] = 675$
$\Rightarrow 3^{2 x} [\frac{25}{27}] = 675$
$\Rightarrow 3^{2 x} = \frac{675 \times 27}{25}$
$\Rightarrow 3^{x} = \sqrt{27 \times 27}$
$\Rightarrow 3^{x} = 27$
$\Rightarrow 3^{x} = 3^{3}$
$x = 3$

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