7.288. $\left\{\begin{array}{l}5^{\sqrt[3]{x}} \cdot 2^{\sqrt{y}}=200 \\ 5^{2 \sqrt[3]{x}}+2^{2 \sqrt{y}}=689\end{array}\right.$

The system of equations is:
\[
\left\{\begin{array}{l}
5^{\sqrt[3]{x}} \cdot 2^{\sqrt{y}}=200 \\
5^{2 \sqrt[3]{x}}+2^{2 \sqrt{y}}=689
\end{array}\right.
\]
## Solution.

Domain of definition: $y \geq 0$.

Rewrite the second equation of the system as

$$
\begin{aligned}
& \left(5^{\sqrt[3]{x}}+2^{\sqrt{y}}\right)^{2}-2 \cdot 5^{\sqrt[3]{x}} \cdot 2^{\sqrt{y}}=689 \Rightarrow\left(5^{\sqrt[3]{x}}+2^{\sqrt{y}}\right)^{2}=1089 \Rightarrow \\
& \Rightarrow 5^{\sqrt[3]{x}}+2^{\sqrt{y}}=-33 \text { (no solutions) or } 5^{\sqrt[3]{x}}+2^{\sqrt{y}}=33
\end{aligned}
$$

Further, we have

$$
\left\{\begin{array}{l}
5^{\sqrt[3]{x}} \cdot 2^{\sqrt{y}}=200 \\
5^{\sqrt[3]{x}}+2^{\sqrt{y}}=33
\end{array} \Rightarrow 2^{\sqrt{y}}=33-5^{\sqrt[3]{x}},\left(5^{\sqrt[3]{x}}\right)^{2}-33 \cdot 5^{\sqrt[3]{x}}+200=0\right.
$$

Solving this equation as a quadratic in terms of $5^{\sqrt[3]{x}}$, we find $5^{\sqrt[3]{x}}=8$, from which $\sqrt[3]{x}=\log _{5} 8, x_{1}=27 \log _{5}^{3} 2$, or $5^{\sqrt[3]{x}}=5^{2}$, from which $\sqrt[3]{x}=2, x_{2}=8$. Then $y_{1}=4 \log _{2}^{2} 5, y_{2}=9$.

Answer: $\quad\left(27 \log _{5}^{3} 2 ; 4 \log _{2}^{2} 5\right),(8 ; 9)$.