Solve the following equations:
$2^{x^{2}} : 2^{2 x} = 8 : 1$ .

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Solution

$2^{x^{2}} : 2^{2 x} = 8 : 1 \frac{2^{x^{2}}}{2^{2 x}} = \frac{8}{1} 2^{x^{2} - 2 x} = 8 2^{x^{2} - 2 x} = 2^{3}$
Comparing both sides
$x^{2} - 2 x = 3 x^{2} - 2 x - 3 = 0 x^{2} - 3 x + x - 3 = 0 x (x - 3) + 1 (x - 3) = 0 (x + 1) (x - 3) = 0 x = - 1, 3$

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