Pembahasan Eksponen UM UGM 2019 Matematika Dasar Kode 633

Soal yang Akan Dibahas
Jika $ \frac{2^{\frac{1}{2}}+2^{\frac{1}{3}}}{2^{-\frac{1}{2}}+2^{-\frac{1}{3}}} = 4^x $, maka $ x = .... $
A). $ \frac{1}{3} \, $ B). $ \frac{5}{12} \, $ C). $ \frac{1}{2} \, $ D). $ \frac{7}{12} \, $ E). $ \frac{2}{3} \, $

$\spadesuit $ Konsep Dasar :
*). Sifat-sifat eksponen :
(i). $ a^{-n} = \frac{1}{a^n} $
(ii). $ a^m. a^n = a^{m+n} $
(iii). $ (a^m)^n = a^{mn} $
*). Persamaan eksponen :
$ a^{f(x)} = a^{g(x)} \rightarrow f(x) = g(x) $
*). Operasi pecahan :
$ \frac{1}{a} + \frac{1}{b} = \frac{a +b}{ab} $
$ a : \frac{b}{c} = a . \frac{c}{b} $

$\clubsuit $ Pembahasan
*). Menentukan nilai $ x $ :
$\begin{align} \frac{2^{\frac{1}{2}}+2^{\frac{1}{3}}}{2^{-\frac{1}{2}}+2^{-\frac{1}{3}}} & = 4^x \\ \frac{2^{\frac{1}{2}}+2^{\frac{1}{3}}}{\frac{1}{2^{\frac{1}{2}}} +\frac{1}{2^{\frac{1}{3}}}} & = 4^x \\ \frac{2^{\frac{1}{2}}+2^{\frac{1}{3}}}{\frac{2^{\frac{1}{2}}+2^{\frac{1}{3}}}{2^{\frac{1}{2}} .2^{\frac{1}{3}}} } & = 4^x \\ (2^{\frac{1}{2}}+2^{\frac{1}{3}}) . \frac{2^{\frac{1}{2}} .2^{\frac{1}{3}}}{2^{\frac{1}{2}}+2^{\frac{1}{3}}} & = 4^x \\ 2^{\frac{1}{2}} .2^{\frac{1}{3}} & = 4^x \\ 2^{\frac{1}{2}+\frac{1}{3}} & = (2^2)^x \\ 2^{\frac{5}{6}} & = 2^{2x} \\ \frac{5}{6} & = 2x \\ x & = \frac{5}{6} . \frac{1}{2} \\ x & = \frac{5}{12} \end{align} $
Jadi, nilai $ x = \frac{5}{12} . \, \heartsuit $

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