20. Show that \( (\sqrt{2}+3 \sqrt{8})^{2}=98 \)

Let's simplify the expression on the left-hand side: \[ (\sqrt{2} + 3\sqrt{8})^2. \] First, notice that \( \sqrt{8} \) can be simplified: \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}. \] So we can rewrite \( 3\sqrt{8} \) as: \[ 3\sqrt{8} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}. \] Now, substitute this back into the original expression: \[ (\sqrt{2} + 6\sqrt{2})^2. \] Combine the like terms: \[ (\sqrt{2} + 6\sqrt{2}) = 7\sqrt{2}. \] Now, we can square this: \[ (7\sqrt{2})^2 = 7^2 \cdot (\sqrt{2})^2 = 49 \cdot 2 = 98. \] Thus, we have shown that: \[ (\sqrt{2}+3 \sqrt{8})^{2} = 98. \] This confirms that the statement is true!