Positive real numbers $r,s$ satisfy the equations $r^2 + s^2 = 1$ and $r^4 + s^4= \frac{7}{8}$.  Find $rs$.
One identity that comes to mind is $(r^2 + s^2)^2 = r^4 + 2r^2s^2 + s^4$. If I plug in the given values of $r^2 + s^2$ and $r^4 + s^4$, I get $(1)^2 = \frac{7}{8} + 2r^2s^2$. Simplifying, I get $2r^2s^2 = \frac{1}{8}$. Dividing both sides by 2, I get $r^2s^2 = \frac{1}{16}$. Taking the square root of both sides, I get $rs = \pm \frac{1}{4}$. However, I know that $r$ and $s$ are positive, so I can ignore the negative sign. Therefore, $rs = \frac{1}{4}$. 