Contoh Soal 3 Fungsi Eksponen (Matematika Peminatan Kelas X)

$\begin{array}{l}\\ 11.& (\mathbf{\text{SPMB 04}})\text{Nilai}x\text{yang memenuhi}\\ & {\displaystyle \frac{27}{3^{2x-1}}={81}^{-0,125}\text{adalah... .}}\\ & \begin{array}{l}\\ \text{a}.& -1{\displaystyle \frac{3}{4}}\\ \text{b}.& -{\displaystyle \frac{3}{4}}\\ \text{c}.& {\displaystyle \frac{3}{4}}\\ \text{d}.& 1{\displaystyle \frac{1}{4}}\\ \text{e}.& 2{\displaystyle \frac{1}{4}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}{\displaystyle \frac{27}{3^{2x-1}}}& ={81}^{-0,125}\\ 3^{3-(2x-1)}& =3^{4(\frac{1}{8})}\\ 3-2x+1& =-{\displaystyle \frac{1}{2}}\\ -2x+4& =-{\displaystyle \frac{1}{2}}\\ -x+2& =-{\displaystyle \frac{1}{4}}\\ -x& =-2-{\displaystyle \frac{1}{4}}\\ -x& =-2{\displaystyle \frac{1}{4}}\\ x& =2{\displaystyle \frac{1}{4}}\end{array}\end{array}$

$\begin{array}{l}\\ 12.& (\mathbf{\text{UMPTN 98}})\text{Bentuk}{\left({\displaystyle \frac{x^{\frac{2}{3}}.y^{\frac{-4}{3}}}{y^{\frac{2}{3}}.x^2}}\right)}^{-\frac{3}{4}}\\ & \text{dapat diserdernakan menjadi}\\ & \begin{array}{l}\\ \text{a}.& \sqrt{x{y}^2}\\ \text{b}.& x\sqrt{y}\\ \text{c}.& \sqrt{x^{2}y}\\ \text{d}.& xy\sqrt{y}\\ \text{e}.& xy\sqrt{x}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}{\left({\displaystyle \frac{x^{\frac{2}{3}}.y^{\frac{-4}{3}}}{y^{\frac{2}{3}}.x^2}}\right)}^{-\frac{3}{4}}& ={(x^{\frac{2}{3}-2}.y^{-\frac{3}{4}-\frac{2}{3}})}^{-\frac{3}{4}}\\ & =x^{-\frac{3}{4}(\frac{2}{3}-2)}.y^{-\frac{3}{4}(-\frac{3}{4}-\frac{2}{3})}\\ & =x^{-\frac{1}{2}+\frac{3}{2}}.y^{1+\frac{1}{2}}\\ & =x^1.y^{1\frac{1}{2}}\\ & =xy\sqrt{y}\end{array}\end{array}$

$\begin{array}{l}\\ 13.& (\mathbf{\text{UMPTN 00}})\\ & \text{Bentuk}{\left(\sqrt[3]{{\displaystyle \frac{1}{243}}}\right)}^{3x}={\left({\displaystyle \frac{3}{3^{x-2}}}\right)}^2\sqrt[3]{{\displaystyle \frac{1}{9}}}\\ & \text{Jika}{x}_0\text{memenuhi persamaan, maka nilai}\\ & 1-{\displaystyle \frac{3}{4}{x}_0=....}\\ & \begin{array}{l}\\ \text{a}.& 1\frac{3}{16}\\ \text{b}.& 1\frac{1}{4}\\ \text{c}.& 1\frac{3}{4}\\ \text{d}.& 2\frac{1}{3}\\ \text{e}.& 2\frac{3}{4}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}{\left(\sqrt[3]{{\displaystyle \frac{1}{243}}}\right)}^{3x}& ={\left({\displaystyle \frac{3}{3^{x-2}}}\right)}^2\sqrt[3]{{\displaystyle \frac{1}{9}}}\\ 3^{-5x}& =3^{2(1-(x-2))}{.3}^{-\frac{2}{3}}\\ -5x& =2(1-(x-2))+(-{\displaystyle \frac{2}{3}}),\text{dikali}3\\ -15x& =6(3-x)+(-2)\\ -15x& =18-6x-2\\ 6x-15x& =16\\ -9x& =16\\ x& ={\displaystyle \frac{16}{-9}}\\ x_0& =-{\displaystyle \frac{16}{9},\text{selanjutnya}}\\ 1-{\displaystyle \frac{3}{4}{x}_0}& =1-{\displaystyle \frac{3}{4}\times (-\frac{16}{9})}\\ & =1+\frac{4}{3}\\ & =1+1{\displaystyle \frac{1}{3}}\\ & =2{\displaystyle \frac{1}{3}}\end{array}\end{array}$

$\begin{array}{l}\\ 14.& \text{Diketahui}{x}^{\frac{1}{2}}+x^{-\frac{1}{2}}=3\\ & \text{Nilai}x+x^{-1}=....\\ & \begin{array}{l}\\ \text{a}.& 7\\ \text{b}.& 8\\ \text{c}.& 8\\ \text{d}.& 10\\ \text{e}.& 11\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}{x}^{\frac{1}{2}}+x^{-\frac{1}{2}}& =3\\ \text{dikadrat}& \text{kan}\\ {(x^{\frac{1}{2}}+x^{-\frac{1}{2}})}^2& =3^2\\ x+2+x^{-1}& =9\\ x+x^{-1}& =9-2\\ & =7\end{array}\end{array}$

$\begin{array}{l}\\ 15.& \text{Diketahui}{2}^{2x}+2^{-2x}=2\\ & \text{Nilai}{2}^x+2^{-x}=....\\ & \begin{array}{l}\\ \text{a}.& 1\\ \text{b}.& 2\\ \text{c}.& \sqrt{2}\\ \text{d}.& 3\\ \text{e}.& \sqrt{3}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}{2}^{2x}+2^{-2x}& =2\\ \text{jika soal}& \text{dikuadratkan}\\ {(2^x+2^{-x})}^2& =2^{2x}+2+2^{-2x}\\ & =2^{2x}+2^{-2x}+2\\ {(2^x+2^{-x})}^2& =2+2=4\\ 2^x+2^{-x}& =\sqrt{4}\\ & =2\end{array}\end{array}$