7. Given the complex number $z=a+b \mathrm{i}(a, b \in \mathbf{Z})$, satisfying $z^{3}=2+11$ i. Then $a+b=$ $\qquad$

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7.3.

From the given conditions,
$$
\begin{array}{l}
|z|^{3}=\sqrt{2^{2}+11^{2}}=\sqrt{125} \\
\Rightarrow|z|=\sqrt{5} \Rightarrow a^{2}+b^{2}=5 \\
\Rightarrow(a, b)=(2,1),(2,-1),(-2,1), \\
(-2,-1) .
\end{array}
$$

Notice that,
$$
(a+b \mathrm{i})^{3}=\left(a^{3}-3 a b^{2}\right)+\left(3 a^{2} b-b^{3}\right) \mathrm{i} \text {. }
$$

Upon verification, only $(a, b)=(2,1)$ satisfies the condition.
Therefore, $a+b=3$.