62nd Ukrainian National Mathematical Olympiad

Let's transform the given expression as follows: $$\begin{gathered} ((7+\sqrt{48}{)}^{2023}+(7-\sqrt{48}{)}^{2023}{)}^2-((7+\sqrt{48}{)}^{2023}-(7-\sqrt{48}{)}^{2023}{)}^2= \\ =((7+\sqrt{48}{)}^{2023}+(7-\sqrt{48}{)}^{2023}+(7+\sqrt{48}{)}^{2023}-(7-\sqrt{48}{)}^{2023})\cdot \\ \cdot ((7+\sqrt{48}{)}^{2023}+(7-\sqrt{48}{)}^{2023}-(7+\sqrt{48}{)}^{2023}+(7-\sqrt{48}{)}^{2023})= \\ =(2(7+\sqrt{48}{)}^{2023})\cdot (2(7-\sqrt{48}{)}^{2023})=2^2(49-48{)}^{2023}=4. \end{gathered}$$ Therefore, the expression equals $4$.