Pembahasan Eksponen UM UGM 2018 Matematika Dasar Kode 585

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

$\spadesuit $ Konsep Dasar :
*). Sifat-sifat perpangkatan/eksponen :
$ ( \sqrt{x} )^2 = x $
$ a^m.a^n = a^{m+n} $
$ (a^m)^n = a^{m.n} $
$ (\frac{a}{b})^n = \frac{a^n}{b^n} $
$ a^{-n} = \frac{1}{a^n} $

$\clubsuit $ Pembahasan
*). Menentukan nilai $ a $ :
$\begin{align} \sqrt{3^{-\frac{1}{2}} + 1} & = \frac{\sqrt{a+1}}{3^{-\frac{1}{4}}} \, \, \, \, \, \, \text{(kuadratkan)} \\ (\sqrt{3^{-\frac{1}{2}} + 1})^2 & = \left( \frac{\sqrt{a+1}}{3^{-\frac{1}{4}}} \right)^2 \\ 3^{-\frac{1}{2}} + 1 & = \frac{(\sqrt{a+1})^2}{(3^{-\frac{1}{4}})^2} \\ 3^{-\frac{1}{2}} + 1 & = \frac{ a+1}{3^{-\frac{1}{2}}} \\ a + 1 & = 3^{-\frac{1}{2}} ( 3^{-\frac{1}{2}} + 1 ) \\ a + 1 & = 3^{-\frac{1}{2}} . 3^{-\frac{1}{2}} + 3^{-\frac{1}{2}} . 1 \\ a + 1 & = 3^{-\frac{1}{2} + (-\frac{1}{2})} + 3^{-\frac{1}{2}} \\ a + 1 & = 3^{-1} + 3^{-\frac{1}{2}} \\ a + 1 & = \frac{1}{3} + 3^{-\frac{1}{2}} \\ a & = \frac{1}{3} - 1 + 3^{-\frac{1}{2}} \\ a & = - \frac{2}{3} + 3^{-\frac{1}{2}} \end{align} $
Jadi, nilai $ a = - \frac{2}{3} + 3^{-\frac{1}{2}} . \, \heartsuit $

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