Question

what is the simplest form of \\(\frac{2\sqrt{2}}{\sqrt{3}-\sqrt{2}}\\)?

\\(2\sqrt{6}+4\\)

\\(2\sqrt{5}+4\\)

\\(\frac{2\sqrt{6}+4}{5}\\)

\\(\frac{2\sqrt{5}+4}{5}\\)

Explanation:

Multiply by the conjugate

Using the Rationalizing the Denominator knowledge point

$$ \frac{2\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{2\sqrt{2}(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})} $$

Simplify the denominator

Using the Rationalizing the Denominator knowledge point

$$ (\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 $$

Expand the numerator

Using the Product Property of Radicals and Distributive Property knowledge points

$$ 2\sqrt{2}(\sqrt{3}+\sqrt{2}) = 2\sqrt{6} + 2(2) = 2\sqrt{6} + 4 $$

Write the final expression

Using the Rationalizing the Denominator knowledge point

$$ \frac{2\sqrt{6} + 4}{1} = 2\sqrt{6} + 4 $$

Answer:

• (A) \(2\sqrt{6}+4\) (Correct answer)
• (B) \(2\sqrt{5}+4\)
• (C) \(\frac{2\sqrt{6}+4}{5}\)
• (D) \(\frac{2\sqrt{5}+4}{5}\)