## Task A-3.2.

For positive real numbers $x, y$ and $z$, if

$$
4^{x+y}=z, \quad z^{1 / x} \cdot z^{1 / y}=1024
$$

determine the value of the expression $\frac{x}{y}+\frac{y}{x}$.
## Solution.

From the second given equation, we have $z^{\frac{1}{x}+\frac{1}{y}}=1024$.

By raising the first given equation to the power of $\frac{1}{x}+\frac{1}{y}$, we get

$$
\begin{aligned}
4^{(x+y)\left(\frac{1}{x}+\frac{1}{y}\right)} & =z^{\frac{1}{x}+\frac{1}{y}}=1024=4^{5}, & & 2 \text { points } \\
(x+y)\left(\frac{1}{x}+\frac{1}{y}\right) & =5, & & 1 \text { point } \\
1+\frac{x}{y}+\frac{y}{x}+1 & =5, & & 1 \text { point } \\
\frac{x}{y}+\frac{y}{x} & =3 . & & 1 \text { point }
\end{aligned}
$$