Pembahasan Bentuk Akar UM UGM 2007 Matematika Dasar

Soal yang Akan Dibahas
$ \frac{5(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})^3}{2\sqrt{2} - \sqrt{3}} = .... $
A). $ \sqrt{3} - \sqrt{2} \, $
B). $ 3\sqrt{3} - 2\sqrt{2} \, $
C). $ 2\sqrt{2} - 3\sqrt{3} $
D). $ 3\sqrt{2} - 2\sqrt{3} $
E). $ 4\sqrt{2} - 3\sqrt{3} $

$\spadesuit $ Konsep Dasar
*). Bentuk perkalian :
$( \sqrt{a} + \sqrt{b})( \sqrt{a} - \sqrt{b}) = a - b $
*). Sifat bentuk akar : $ \sqrt{a.b} = \sqrt{a}. \sqrt{b} $

$\clubsuit $ Pembahasan
*). Menyelesaikan soal :
$\begin{align} & \frac{5(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})^3}{2\sqrt{2} - \sqrt{3}} \\ & = \frac{5(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) (\sqrt{3} - \sqrt{2})^2}{2\sqrt{2} - \sqrt{3}} \\ & = \frac{5(3 - 2) (\sqrt{3} - \sqrt{2})^2}{2\sqrt{2} - \sqrt{3}} \\ & = \frac{5(\sqrt{3} - \sqrt{2})^2}{2\sqrt{2} - \sqrt{3}} \\ & = \frac{5(3 - 2\sqrt{6} + 2)}{2\sqrt{2} - \sqrt{3}} \\ & = \frac{5(5 - 2\sqrt{6})}{2\sqrt{2} - \sqrt{3}} \\ & = \frac{5(5 - 2\sqrt{6})}{2\sqrt{2} - \sqrt{3}} . \frac{2\sqrt{2} + \sqrt{3}}{2\sqrt{2} + \sqrt{3}} \\ & = \frac{5( 10\sqrt{2} + 5\sqrt{3} - 4\sqrt{12} - 2\sqrt{18})}{(2\sqrt{2})^2 - (\sqrt{3})^2} \\ & = \frac{5( 10\sqrt{2} + 5\sqrt{3} - 4. \sqrt{4}.\sqrt{3} - 2.\sqrt{9}.\sqrt{2})}{(2\sqrt{2})^2 - (\sqrt{3})^2} \\ & = \frac{5( 10\sqrt{2} + 5\sqrt{3} - 4. 2 \sqrt{3} - 2. 3\sqrt{2})}{8 - 3} \\ & = \frac{5( 10\sqrt{2} + 5\sqrt{3} - 8 \sqrt{3} - 6\sqrt{2})}{5} \\ & = 4\sqrt{2} - 3\sqrt{3} \end{align} $
Jadi, hasilnya adalah $ 4\sqrt{2} - 3\sqrt{3} . \, \heartsuit $

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