Question

rationalize the denominator. simplify if possible.

\\\frac{\sqrt{2} - 4}{\sqrt{18} - \sqrt{2}}\\

\\\frac{\sqrt{2} - 4}{\sqrt{18} - \sqrt{2}} = \square\\

(simplify your answer. type an exact answer using radicals as needed.)

Explanation:

⚡ Using what you learned: Rationalizing Denominators

Step 1: Simplify the denominator

$$ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} $$

$$ \sqrt{18} - \sqrt{2} = 3\sqrt{2} - \sqrt{2} = 2\sqrt{2} $$

Step 2: Rewrite the expression

$$ \frac{\sqrt{2} - 4}{2\sqrt{2}} $$

Step 3: Rationalize the denominator

Multiply the numerator and the denominator by \(\sqrt{2}\):

$$ \frac{(\sqrt{2} - 4) \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = \frac{2 - 4\sqrt{2}}{2 \cdot 2} = \frac{2 - 4\sqrt{2}}{4} $$

Step 4: Simplify the fraction

Divide each term in the numerator by \(2\):

$$ \frac{2(1 - 2\sqrt{2})}{4} = \frac{1 - 2\sqrt{2}}{2} $$

Answer:

$$ \frac{1 - 2\sqrt{2}}{2} $$