Simplify the square roots in the numerator and denominator.

Let's simplify this expression. Step 1: Simplify the square roots in the numerator and denominator. $$ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} $$ $$ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} $$ Step 2: Substitute the simplified square roots back into the expression. $$ \frac{8 - 4\sqrt{18}}{\sqrt{50}} = \frac{8 - 4(3\sqrt{2})}{5\sqrt{2}} $$ $$ = \frac{8 - 12\sqrt{2}}{5\sqrt{2}} $$ Step 3: Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$. $$ \frac{8 - 12\sqrt{2}}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} $$ $$ = \frac{(8 - 12\sqrt{2})\sqrt{2}}{5\sqrt{2}\sqrt{2}} $$ $$ = \frac{8\sqrt{2} - 12(\sqrt{2})^2}{5(2)} $$ $$ = \frac{8\sqrt{2} - 12(2)}{10} $$ $$ = \frac{8\sqrt{2} - 24}{10} $$ Step 4: Separate the terms and simplify the fraction. $$ = \frac{8\sqrt{2}}{10} - \frac{24}{10} $$ $$ = \frac{4\sqrt{2}}{5} - \frac{12}{5} $$ Step 5: Rearrange the terms to match the form $a + b\sqrt{n}$. $$ = -\frac{12}{5} + \frac{4}{5}\sqrt{2} $$ Step 6: Identify the values of $a$, $b$, and $n$. Comparing $-\frac{12}{5} + \frac{4}{5}\sqrt{2}$ with $a + b\sqrt{n}$: $a = -\frac{12}{5}$ $b = \frac{4}{5}$ $n = 2$ The simplified expression is $\boxed{\text{$-\frac{12}{5} + \frac{4}{5}\sqrt{2}$}}$. The values are $\boxed{\text{$a = -\frac{12}{5}$, $b = \frac{4}{5}$, $n = 2$}}$. 3 done, 2 left today. You're making progress.