Soal yang Akan Dibahas
Bentuk sederhana dari $ 78 \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) $
adalah ...
A). $ 234 \, $ B). $ 312 \, $ C). $ 468 $ D). $ 546 $ E). $ 624 $
A). $ 234 \, $ B). $ 312 \, $ C). $ 468 $ D). $ 546 $ E). $ 624 $
$\spadesuit $ Konsep Dasar :
*). Sifat eksponen dan bentuk akar :
$ a^2 = b \rightarrow a = b^\frac{1}{2} = \sqrt{b} $
$ (\sqrt{a})^2 = a $
$ (a+ b)^2 = a^2 + b^2 + 2ab $
$ \sqrt{a}.\sqrt{b} = \sqrt{a.b} $
$ (a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2.c $
*). Sifat eksponen dan bentuk akar :
$ a^2 = b \rightarrow a = b^\frac{1}{2} = \sqrt{b} $
$ (\sqrt{a})^2 = a $
$ (a+ b)^2 = a^2 + b^2 + 2ab $
$ \sqrt{a}.\sqrt{b} = \sqrt{a.b} $
$ (a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2.c $
$\clubsuit $ Pembahasan
*). Menentukan hasil $ \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) $ :
misalkan $ t = \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) $
$ \begin{align} t^2 & = \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right)^2 \\ & = \sqrt{17+12\sqrt{2}} ^2 + \sqrt{17-12\sqrt{2}} ^2 + 2 \sqrt{17+12\sqrt{2}} . \sqrt{17-12\sqrt{2}} \\ & = 17+12\sqrt{2} + 17-12\sqrt{2} + 2 \sqrt{(17+12\sqrt{2}).(17-12\sqrt{2})} \\ & = 34 + 2 \sqrt{17^2 - 12^2.2} \\ & = 34 + 2 \sqrt{289 - 288} \\ & = 34 + 2 \sqrt{1} \\ & = 34 + 2 .1 \\ t^2 & = 36 \\ t & = \sqrt{36} \\ t & = 6 \end{align} $
sehingga nilai $ \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) = t = 6 $
*). Hasil akhirnya :
$ \begin{align} 78 \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) & = 78 \times 6 = 468 \end{align} $
Jadi, nilai $ 78\left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) = 468 . \, \heartsuit $
*). Menentukan hasil $ \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) $ :
misalkan $ t = \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) $
$ \begin{align} t^2 & = \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right)^2 \\ & = \sqrt{17+12\sqrt{2}} ^2 + \sqrt{17-12\sqrt{2}} ^2 + 2 \sqrt{17+12\sqrt{2}} . \sqrt{17-12\sqrt{2}} \\ & = 17+12\sqrt{2} + 17-12\sqrt{2} + 2 \sqrt{(17+12\sqrt{2}).(17-12\sqrt{2})} \\ & = 34 + 2 \sqrt{17^2 - 12^2.2} \\ & = 34 + 2 \sqrt{289 - 288} \\ & = 34 + 2 \sqrt{1} \\ & = 34 + 2 .1 \\ t^2 & = 36 \\ t & = \sqrt{36} \\ t & = 6 \end{align} $
sehingga nilai $ \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) = t = 6 $
*). Hasil akhirnya :
$ \begin{align} 78 \left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) & = 78 \times 6 = 468 \end{align} $
Jadi, nilai $ 78\left( \sqrt{17+12\sqrt{2}} + \sqrt{17-12\sqrt{2}} \right) = 468 . \, \heartsuit $
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