Question 1209872: Simplify \frac{6 \sqrt{2]}{\sqrt{2} - \sqrt{3} - \sqrt{5}}
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To simplify the given expression, we need to rationalize the denominator.
Let the expression be denoted by $E$:
$$E = \frac{6\sqrt{2}}{\sqrt{2} - \sqrt{3} - \sqrt{5}}$$
First, multiply the numerator and denominator by $(\sqrt{2} - \sqrt{3} + \sqrt{5})$:
$$E = \frac{6\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})}{(\sqrt{2} - \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} + \sqrt{5})}$$
The denominator is of the form $(a-b)(a+b) = a^2 - b^2$, where $a = \sqrt{2} - \sqrt{3}$ and $b = \sqrt{5}$.
So, the denominator becomes:
$$(\sqrt{2} - \sqrt{3})^2 - (\sqrt{5})^2 = (2 - 2\sqrt{6} + 3) - 5 = 5 - 2\sqrt{6} - 5 = -2\sqrt{6}$$
Thus,
$$E = \frac{6\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})}{-2\sqrt{6}} = \frac{3\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})}{-\sqrt{6}}$$
Multiply the numerator and denominator by $\sqrt{6}$:
$$E = \frac{3\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})(-\sqrt{6})}{6} = -\frac{\sqrt{2}(\sqrt{2} - \sqrt{3} + \sqrt{5})\sqrt{6}}{2}$$
$$E = -\frac{\sqrt{12} - \sqrt{18} + \sqrt{30}}{2} = -\frac{2\sqrt{3} - 3\sqrt{2} + \sqrt{30}}{2}$$
Therefore,
$$E = \frac{3\sqrt{2} - 2\sqrt{3} - \sqrt{30}}{-2} = \frac{-3\sqrt{2} + 2\sqrt{3} + \sqrt{30}}{2}$$
Thus,
$$E = \frac{2\sqrt{3} - 3\sqrt{2} + \sqrt{30}}{2}$$
Final Answer: The final answer is $\boxed{\frac{2\sqrt{3} - 3\sqrt{2} + \sqrt{30}}{2}}$
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