7. Let integers $a, b$ satisfy
$$
3 \sqrt{\sqrt[3]{5}-\sqrt[3]{4}}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{2} \text {. }
$$

Then $a b=$ $\qquad$
7. -500 .

From the expansion of $(\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{2})^{2}$, after combining like terms, only $\sqrt[3]{5} \sqrt[3]{4}$ remains. Considering that $a$ and $b$ contain only the factors 2 and 5, we have:
$$
a=20, b=-25 \text { or } a=-25, b=20 \text {. }
$$

Thus, $a b=-500$.