7. (1) Does there exist a cube that can be expressed as the sum of two squares?
(2) Does there exist a fourth power that can be expressed as the sum of two squares?
7. (1) There exists.

For example, $10^{2}+5^{2}=5^{3}$.
(2) There exists.

For example, $15^{2}+20^{2}=5^{4}$.