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SMUP 2010

Solusi SMUP 2010 Kode 041 – Nomor 7


Jika\ 4^x=6,\ maka\ \frac{8^x+8^{-x}}{2^x+2^{-x}}=....\\\\A.\quad  \frac{29}{8}\\\\B.\quad \frac{143}{40}\\\\C.\quad \frac{31}{6}\\\\D.\quad \frac{16}{3}\\\\E.\quad \frac{43}{6}
Jawaban: C

\begin{array} {rcl} 4^x&=&6\\\big(2^x\big)^2&=&6\\2^x&=&\sqrt{6}\end{array}
sehingga diperoleh:

\begin{array} {rcl} \frac{8^x+8^{-x}}{2^x+2^{-x}}&=&\frac{\big(2^x\big)^3+\big(2^x\big)^{-3}}{2^x+2^{-x}}\\\\&=&\frac{\big(\sqrt{6}\big)^3+\big(\sqrt{6}\big)^{-3}}{\sqrt{6}+\big(\sqrt{6}\big)^{-1}}\\\\&=&\frac{6\sqrt{6}+\frac{1}{6\sqrt{6}}}{\sqrt{6}+\frac{1}{\sqrt{6}}}\quad \times \quad \frac{\sqrt{6}}{\sqrt{6}}\\\\&=&\frac{36+\frac{1}{6}}{6+1}\\\\&=&\frac{\frac{217}{6}}{7}\\\\&=&\frac{31}{6}\end{array}

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