Simplify \(\frac{1 + \sqrt{8}}{3 - \sqrt{2}}\).
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Correct Answer: Option D
Explanation:
\(\frac{1 + \sqrt{8}}{3 - \sqrt{2}}\)
Rationalizing by multiplying through with \(3 + \sqrt{2}\),
\((\frac{1 + \sqrt{8}}{3 - \sqrt{2}})(\frac{3 + \sqrt{2}}{3 + \sqrt{2}}) = \frac{3 + \sqrt{2} + 3\sqrt{8} + 4}{9 - 2}\)
= \(\frac{3 + \sqrt{2} + 3\sqrt{4 \times 2} + 4}{7} \)
= \(\frac{7 + 7\sqrt{2}}{7} = 1 + \sqrt{2}\)
\(\frac{1 + \sqrt{8}}{3 - \sqrt{2}}\)
Rationalizing by multiplying through with \(3 + \sqrt{2}\),
\((\frac{1 + \sqrt{8}}{3 - \sqrt{2}})(\frac{3 + \sqrt{2}}{3 + \sqrt{2}}) = \frac{3 + \sqrt{2} + 3\sqrt{8} + 4}{9 - 2}\)
= \(\frac{3 + \sqrt{2} + 3\sqrt{4 \times 2} + 4}{7} \)
= \(\frac{7 + 7\sqrt{2}}{7} = 1 + \sqrt{2}\)