3. Real numbers $a$ and $b$ satisfy these equations:

$$
a^{3}-3 a b^{2}=44, \quad b^{3}-3 a^{2} b=8
$$

What is $a^{2}+b^{2}$?
3. We have

$$
\left(a^{3}-3 a b^{2}\right)^{2}+\left(b^{3}-3 a^{2} b\right)^{2}=44^{2}+8^{2}=2000
$$

On the other hand,

$$
\begin{aligned}
\left(a^{3}-3 a b^{2}\right)^{2}+\left(b^{3}-3 a^{2} b\right)^{2}= & a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4} \\
& +b^{6}-6 a^{2} b^{4}+9 a^{2} b^{4} \\
= & a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6} \\
= & \left(a^{2}+b^{2}\right)^{3}
\end{aligned}
$$

From here, we get $\left(a^{2}+b^{2}\right)^{3}=2000$, i.e., $a^{2}+b^{2}=10 \sqrt[3]{2}$.

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