Ask Uncle Colin: Powers

Dear Uncle Colin,

I need to solve $5^{2 x + 2} + 16 \cdot 15^{x} - 9^{x + 1} = 0$ but I’ve hit a dead end! Can you help?

- Puzzling Out Wild Exponent Relations

Hi, POWER, and thanks for your message!

From the working you sent, it looks like you’ve picked a good strategy: you’ve let $X = 5^{x}$ and $Y = 3^{x}$ to make things a bit tidier, turning the whole thing into a quadratic in two variables.

That gives you $25 X^{2} + 16 X Y - 9 Y^{2} = 0$ , which looks like a mess.

Fortunately, it factorises as $(25 X - 9 Y) (X + Y) = 0$ , so either $\frac{Y}{X} = \frac{25}{9}$ or $X = - Y$ .

The last of those is impossible (exponentials are always positive), so it must be the case that $\frac{3^{x}}{5^{x}} = \frac{25}{9}$

There are several ways to tackle this; by inspection, $x = - 2$ works. Alternatively, you can take logs to say $x \ln (\frac{3}{5}) = \ln (\frac{25}{9})$ and let your calculator do the not-exactly-heavy lifting.

Hope that helps!

- Uncle Colin

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