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Simplify the square roots in the numerator and denominator.

Mathematics

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Let's simplify this expression.

  1. Simplify the square roots in the numerator and denominator.

    18=9×2=32\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} 50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
  2. Substitute the simplified square roots back into the expression.

    8−41850=8−4(32)52\frac{8 - 4\sqrt{18}}{\sqrt{50}} = \frac{8 - 4(3\sqrt{2})}{5\sqrt{2}} =8−12252= \frac{8 - 12\sqrt{2}}{5\sqrt{2}}
  3. Rationalize the denominator by multiplying the numerator and denominator by 2\sqrt{2}.

    8−12252×22\frac{8 - 12\sqrt{2}}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} =(8−122)2522= \frac{(8 - 12\sqrt{2})\sqrt{2}}{5\sqrt{2}\sqrt{2}} =82−12(2)25(2)= \frac{8\sqrt{2} - 12(\sqrt{2})^2}{5(2)} =82−12(2)10= \frac{8\sqrt{2} - 12(2)}{10} =82−2410= \frac{8\sqrt{2} - 24}{10}
  4. Separate the terms and simplify the fraction.

    =8210−2410= \frac{8\sqrt{2}}{10} - \frac{24}{10} =425−125= \frac{4\sqrt{2}}{5} - \frac{12}{5}
  5. Rearrange the terms to match the form a+bna + b\sqrt{n}.

    =−125+452= -\frac{12}{5} + \frac{4}{5}\sqrt{2}
  6. Identify the values of aa, bb, and nn.

    Comparing −125+452-\frac{12}{5} + \frac{4}{5}\sqrt{2} with a+bna + b\sqrt{n}:
    a=−125a = -\frac{12}{5}
    b=45b = \frac{4}{5}
    n=2n = 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}}.

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