Pembahasan Eksponen UM UGM 2006 Matematika Dasar

Soal yang Akan Dibahas
Bentuk sederhana dari
$ \frac{\left( x^{-4}y^\frac{2}{3} \right)^{-\frac{1}{2}} \left( x^{-\frac{7}{3}} y^{-1} \right)^\frac{1}{2} }{\left( x^\frac{1}{2} y^3\right)^{-\frac{1}{6}} \left(x^{-\frac{1}{4}} y^{-1} \right)^\frac{1}{3} } $
adalah ....
A). $ y \, $ B). $ x \, $ C). $ xy \, $ D). $ \frac{x}{y} \, $ E). $ \frac{y}{x} \, $

$\spadesuit $ Konsep Dasar
*). Sifat-sifat Eksponen :
$ a^m.a^n = a^{m+n} \, $ , $ (a^m)^n = a^{m.n} $
$ (a.b)^n = a^n.b^n \, $ , $ \frac{a^m}{a^n} = a^{m-n} $

$\clubsuit $ Pembahasan
*). Menyelesaikan Soal :
$\begin{align} & \frac{\left( x^{-4}y^\frac{2}{3} \right)^{-\frac{1}{2}} \left( x^{-\frac{7}{3}} y^{-1} \right)^\frac{1}{2} }{\left( x^\frac{1}{2} y^3\right)^{-\frac{1}{6}} \left(x^{-\frac{1}{4}} y^{-1} \right)^\frac{1}{3} } \\ & = \frac{\left( x^{-4}\right)^{-\frac{1}{2}} \left( y^\frac{2}{3} \right)^{-\frac{1}{2}} \left( x^{-\frac{7}{3}} \right)^\frac{1}{2} \left( y^{-1} \right)^\frac{1}{2} }{\left( x^\frac{1}{2} \right)^{-\frac{1}{6}} \left( y^3\right)^{-\frac{1}{6}} \left(x^{-\frac{1}{4}} \right)^\frac{1}{3} \left( y^{-1} \right)^\frac{1}{3} } \\ & = \frac{x^2. y^{-\frac{1}{3}} . x^{-\frac{7}{6}} . y^{-\frac{1}{2}} }{ x^{-\frac{1}{12}} .y^{-\frac{1}{2}} .x^{-\frac{1}{12}} . y^{-\frac{1}{3}} } \\ & = \frac{x^2. x^{-\frac{7}{6}} }{ x^{-\frac{1}{12}} .x^{-\frac{1}{12}} } = \frac{ x^{2 -\frac{7}{6}} }{ x^{-\frac{1}{12} - \frac{1}{12}} } = \frac{ x^{\frac{5}{6}} }{ x^{-\frac{1}{6}} } \\ & = x^{\frac{5}{6} - (-\frac{1}{6})} = x^{\frac{6}{6}} = x^1 = x \end{align} $
Jadi, bentuk sederhananya adalah $ x \, \heartsuit $

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