Question 7

If $$(\sqrt{2})^{\sqrt{x}} - (\sqrt{2})^{\sqrt{y}} = 48$$, where $$x$$ and $$y$$ are natural numbers, then the value of $$x + y$$ is

Solution

The equation is $$2^{\frac{\sqrt{x}}{2}} - 2^{\frac{\sqrt{y}}{2}} = 48$$, and writing $$48 = 2^4(2^2 - 1)$$ shows the only way to express 48 as a difference of two powers of 2 is $$2^6 - 2^4$$. Hence $$\frac{\sqrt{x}}{2} = 6$$ and $$\frac{\sqrt{y}}{2} = 4$$, giving $$\sqrt{x} = 12$$ and $$\sqrt{y} = 8$$, so $$x = 144$$ and $$y = 64$$. Therefore $$x + y = 208$$.

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