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Simplify \(\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}\)


Question

Simplify \(\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}\)

Options

A) 3\(\sqrt{6} - 7\)

B) 3\(\sqrt{6} - 7\)

C) 3\(\sqrt{6} - 1\)

D) 3\(\sqrt{6} + 1\)

The correct answer is A.

Explanation:

\(= \frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}\)
\(= \frac{2\sqrt{2}(\sqrt{2}) + (2\sqrt{2})(-\sqrt{3})-\sqrt{3}(\sqrt{2})-\sqrt{3}(-\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}\)
\(= \frac{2 \times 2 - 2\sqrt{6} - \sqrt{6} + 3}{2 - 3}\)
\(= \frac{4 - 3\sqrt{6} + 3}{-1}\)
\(= \frac{7 - 3\sqrt{6}}{-1}\)
\(= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}\)
\(= -7 + 3\sqrt{6}\)
\(= 3\sqrt{6}-7\)

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Dicussion (1)

  • \(= \frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}\)
    \(= \frac{2\sqrt{2}(\sqrt{2}) + (2\sqrt{2})(-\sqrt{3})-\sqrt{3}(\sqrt{2})-\sqrt{3}(-\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}\)
    \(= \frac{2 \times 2 - 2\sqrt{6} - \sqrt{6} + 3}{2 - 3}\)
    \(= \frac{4 - 3\sqrt{6} + 3}{-1}\)
    \(= \frac{7 - 3\sqrt{6}}{-1}\)
    \(= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}\)
    \(= -7 + 3\sqrt{6}\)
    \(= 3\sqrt{6}-7\)

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