Contoh Soal 3 Fungsi Eksponen (Matematika Peminatan Kelas X)

$\begin{array}{ll}\\ 11.&(\textbf{SPMB 04})\textrm{Nilai}\: \: x\: \: \textrm{yang memenuhi}\\ &\displaystyle \frac{27}{3^{2x-1}}=81^{-0,125} \: \: \textrm{adalah... .}\\ &\begin{array}{llll}\\ \textrm{a}.&-1\displaystyle \frac{3}{4}\\ \textrm{b}.&-\displaystyle \frac{3}{4}\\ \textrm{c}.&\displaystyle \frac{3}{4}\\ \textrm{d}.&1\displaystyle \frac{1}{4}\\ \color{red}\textrm{e}.&2\displaystyle \frac{1}{4} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}\displaystyle \frac{27}{3^{2x-1}}&=81^{-0,125}\\ 3^{3-(2x-1)}&=3^{4(\frac{1}{8})}\\ 3-2x+1&=-\displaystyle \frac{1}{2}\\ -2x+4&=-\displaystyle \frac{1}{2}\\ -x+2&=-\displaystyle \frac{1}{4}\\ -x&=-2-\displaystyle \frac{1}{4}\\ -x&=-2\displaystyle \frac{1}{4}\\ x&=2\displaystyle \frac{1}{4} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 12.&(\textbf{UMPTN 98})\textrm{Bentuk}\: \: \left ( \displaystyle \frac{x^{\frac{2}{3}}.y^{\frac{-4}{3}}}{y^{\frac{2}{3}}.x^{2}} \right )^{-\frac{3}{4}}\\ &\textrm{dapat diserdernakan menjadi}\\ &\begin{array}{llll}\\ \textrm{a}.&\sqrt{xy^{2}}\\ \textrm{b}.&x\sqrt{y}\\ \textrm{c}.&\sqrt{x^{2}y}\\ \color{red}\textrm{d}.&xy\sqrt{y}\\ \textrm{e}.&xy\sqrt{x} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\left ( \displaystyle \frac{x^{\frac{2}{3}}.y^{\frac{-4}{3}}}{y^{\frac{2}{3}}.x^{2}} \right )^{-\frac{3}{4}}&=\left ( x^{\frac{2}{3}-2}.y^{-\frac{3}{4}-\frac{2}{3}} \right )^{-\frac{3}{4}}\\ &=x^{-\frac{3}{4}(\frac{2}{3}-2)}.y^{-\frac{3}{4}(-\frac{3}{4}-\frac{2}{3})}\\ &=x^{-\frac{1}{2}+\frac{3}{2}}.y^{1+\frac{1}{2}}\\ &=x^{1}.y^{1\frac{1}{2}}\\ &=xy\sqrt{y} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 13.&(\textbf{UMPTN 00})\\ &\textrm{Bentuk}\: \: \left (\sqrt[3]{\displaystyle \frac{1}{243}} \right )^{3x}=\left ( \displaystyle \frac{3}{3^{x-2}} \right )^{2}\sqrt[3]{\displaystyle \frac{1}{9}}\\ &\textrm{Jika}\: \: x_{0}\: \: \textrm{memenuhi persamaan, maka nilai}\\ &1-\displaystyle \frac{3}{4}x_{0}=\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&1\frac{3}{16}\\ \textrm{b}.&1\frac{1}{4}\\ \textrm{c}.&1\frac{3}{4}\\ \color{red}\textrm{d}.&2\frac{1}{3}\\ \textrm{e}.&2\frac{3}{4} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{d}\\ &\color{blue}\begin{aligned}\left (\sqrt[3]{\displaystyle \frac{1}{243}} \right )^{3x}&=\left ( \displaystyle \frac{3}{3^{x-2}} \right )^{2}\sqrt[3]{\displaystyle \frac{1}{9}}\\ 3^{-5x}&=3^{2(1-(x-2))}.3^{-\frac{2}{3}}\\ -5x&=2(1-(x-2))+\left ( -\displaystyle \frac{2}{3} \right ),\: \: \textrm{dikali}\: \: 3\\ -15x&=6(3-x)+(-2)\\ -15x&=18-6x-2\\ 6x-15x&=16\\ -9x&=16\\ x&=\displaystyle \frac{16}{-9}\\ x_{0}&=-\displaystyle \frac{16}{9},\: \: \textrm{selanjutnya}\\ 1-\displaystyle \frac{3}{4}x_{0}&=1-\displaystyle \frac{3}{4}\times \left (-\frac{16}{9} \right )\\ &=1+\frac{4}{3}\\ &=1+1\displaystyle \frac{1}{3}\\ &=2\displaystyle \frac{1}{3} \end{aligned} \end{array}$

$\begin{array}{ll}\\ 14.&\textrm{Diketahui}\: \: x^{\frac{1}{2}}+x^{-\frac{1}{2}}=3\\ &\textrm{Nilai}\: \: x+x^{-1}=....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&7\\ \textrm{b}.&8\\ \textrm{c}.&8\\ \textrm{d}.&10\\ \textrm{e}.&11 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}x^{\frac{1}{2}}+x^{-\frac{1}{2}}&=3\\ \textrm{dikadrat}&\textrm{kan}\\ \left ( x^{\frac{1}{2}}+x^{-\frac{1}{2}} \right )^{2}&=3^{2}\\ x+2+x^{-1}&=9\\ x+x^{-1}&=9-2\\ &=7 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 15.&\textrm{Diketahui}\: \: 2^{2x}+2^{-2x}=2\\ &\textrm{Nilai}\: \: 2^{x}+2^{-x}=....\\ &\begin{array}{llll}\\ \textrm{a}.&1\\ \color{red}\textrm{b}.&2\\ \textrm{c}.&\sqrt{2}\\ \textrm{d}.&3\\ \textrm{e}.&\sqrt{3} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}2^{2x}+2^{-2x}&=2\\ \textrm{jika soal}&\: \textrm{dikuadratkan}\\ \left ( 2^{x}+2^{-x} \right )^{2}&=2^{2x}+2+2^{-2x}\\ &=2^{2x}+2^{-2x}+2\\ \left ( 2^{x}+2^{-x} \right )^{2}&=2+2=4\\ 2^{x}+2^{-x}&=\sqrt{4}\\ &=2 \end{aligned} \end{array}$