Lanjutan 1 Contoh Soal dan Pembahasan Persiapan PHB Gasal Materi Fungsi Eksponensial (Kelas X)

$\begin{array}{l}\\ 6.& \text{Bentuk sederhana dari}\\ & {\left({\displaystyle \frac{1}{2}}\right)}^{-{\left(\frac{1}{2}\right)}^{-1}}+{\left({\displaystyle \frac{1}{3}}\right)}^{-{\left(\frac{1}{3}\right)}^{-1}}+{\left({\displaystyle \frac{1}{4}}\right)}^{-{\left(\frac{1}{4}\right)}^{-1}}+{\left({\displaystyle \frac{1}{5}}\right)}^{-{\left(\frac{1}{5}\right)}^{-1}}\\ & \text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 3142\\ \text{b}.& 287\\ \text{c}.& 3412\\ \text{d}.& 4116\\ \text{e}.& 4096\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& {\left({\displaystyle \frac{1}{2}}\right)}^{-{\left(\frac{1}{2}\right)}^{-1}}+{\left({\displaystyle \frac{1}{3}}\right)}^{-{\left(\frac{1}{3}\right)}^{-1}}+{\left({\displaystyle \frac{1}{4}}\right)}^{-{\left(\frac{1}{4}\right)}^{-1}}+{\left({\displaystyle \frac{1}{5}}\right)}^{-{\left(\frac{1}{5}\right)}^{-1}}\\ & ={\left({\displaystyle \frac{1}{2}}\right)}^{-2}+{\left({\displaystyle \frac{1}{3}}\right)}^{-3}+{\left({\displaystyle \frac{1}{4}}\right)}^{-4}+{\left({\displaystyle \frac{1}{5}}\right)}^{-5}\\ & =2^2+3^3+4^4+5^5\\ & =4+27+256+3125\\ & =3412\end{array}\end{array}$.

$\begin{array}{l}\\ 7.& \text{Nilai dari}\\ & \underset{89}{\underbrace{(-7)(-7)(-7)\cdots (-7)}}-(-7{)}^{89}\\ & \text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 2\\ \text{b}.& 7\\ \text{c}.& 4\\ \text{d}.& 0\\ \text{e}.& 9\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}& \underset{89}{\underbrace{(-7)(-7)(-7)\cdots (-7)}}-(-7{)}^{89}\\ & =(-7{)}^{89}-(-7{)}^{89}\\ & =0\end{array}\end{array}$.

$\begin{array}{l}\\ 8.& \text{Nilai dari}{\left(\sqrt{3\times \sqrt[4]{27\times \sqrt[3]{81}}}\right)}^{{\displaystyle \frac{24}{25}}}\\ & \text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 1\\ \text{b}.& 2\\ \text{c}.& 3\\ \text{d}.& 4\\ \text{e}.& 5\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& {\left(\sqrt{3\times \sqrt[4]{27\times \sqrt[3]{81}}}\right)}^{{\displaystyle \frac{24}{25}}}\\ & ={(3^{\frac{1}{2}}.(3^3{)}^{\frac{1}{2.4}}.(3^4{)}^{\frac{1}{2.4.3}})}^{{\displaystyle \frac{24}{25}}}\\ & ={\left(3^{\frac{1}{2}+\frac{3}{8}+\frac{4}{24}}\right)}^{{\displaystyle \frac{24}{25}}}\\ & =3^{\frac{12}{25}+\frac{9}{25}+\frac{4}{25}}\\ & =3^{\frac{25}{25}}\\ & =3\end{array}\end{array}$.

$\begin{array}{l}\\ 9.& \text{Jika}\sqrt{{27}^{2x+3}}={\displaystyle \frac{1}{3^{x-2}{.9}^{3x}},\text{maka nilai}}\\ & 8x+2\text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 0\\ \text{b}.& 2\\ \text{c}.& 4\\ \text{d}.& 8\\ \text{e}.& 12\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \mathbf{\text{Alternatif 1}}\\ & \begin{array}{rl}\sqrt{{27}^{2x+3}}& ={\displaystyle \frac{1}{3^{x-2}{.9}^{3x}}}\\ {27}^{{\displaystyle \frac{2x+3}{2}}}& =3^{-(x-2)}{.9}^{-3x}\\ 3^{3.\left({\displaystyle \frac{2x+3}{2}}\right)}& =3^{2-x}{.3}^{2(-3x)}\\ 3^{3.\left({\displaystyle \frac{2x+3}{2}}\right)}& =3^{2-x-6x}\\ 3{}^{3.\left({\displaystyle \frac{2x+3}{2}}\right)}& =3{}^{2-7x}\\ 3\frac{(2x+3)}{2}& =2-7x\\ 6x+9& =4-14x\\ 6x+14x& =4-9\\ 20x& =-5\\ & x=-{\displaystyle \frac{1}{4}}\\ \text{maka nilai}& \\ 8x+2& =8(-{\displaystyle \frac{1}{4}})+2=-2+2=0\end{array}\\ & \mathbf{\text{Alternatif 2}}\\ & \begin{array}{rl}\sqrt{{27}^{2x+3}}& ={\displaystyle \frac{1}{3^{x-2}{.9}^{3x}}}\\ {27}^{{\displaystyle \frac{2x+3}{2}}}{.3}^{x-2}{.9}^{3x}& =1\\ 3^{3\left({\displaystyle \frac{2x+3}{2}}\right)}{.3}^{x-2}{.3}^{2(3x)}& =3^0\\ 3^{3\left({\displaystyle \frac{2x+3}{2}}\right)+x-2+6x}& =3^0\\ {\displaystyle \frac{3(2x+3)}{2}+7x-2}& =0\\ 6x+9+14x-4& =0\\ 20x+5& =0\\ 20x& =-5\\ x& =-{\displaystyle \frac{1}{4}}\\ \text{Selanjutnya sama}& \\ \text{dengan langkah no.1}& \text{di atas}\end{array}\end{array}$.

$\begin{array}{l}\\ 10.& \text{Penyelesaian persamaan}{3}^{2x+1}={81}^{x-2}\\ & \text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 0\\ \text{b}.& 2\\ \text{c}.& 4\\ \text{d}.& 4{\displaystyle \frac{1}{2}}\\ \text{e}.& 16\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}{3}^{2x+1}& ={81}^{x-2}\\ 3^{2x+1}& =(3^4{)}^{x-2}\\ 3^{2x+1}& =3^{4x-8}\\ 2x+1& =4x-8\\ 2x-4x& =-8-1\\ -2x& =-9\\ x& ={\displaystyle \frac{-9}{-2}}\\ & =4{\displaystyle \frac{1}{2}}\end{array}\end{array}$.

Contoh Soal dan Pembahasan Persiapan PHB Gasal Materi Fungsi Eksponensial (Kelas X)

 $\begin{array}{l}\\ 1.& \text{Bentuk sederhana dari}{\displaystyle \frac{a^p.a^q}{a^r}:a^{2r}\text{adalah ... .}}\\ & \begin{array}{l}\\ \text{a}.& a^{p+q-3r}\\ \text{b}.& a^{p+3r-q}\\ \text{c}.& a^{p-2q+r}\\ \text{d}.& a^{p+q+r}\\ \text{e}.& a^{p-3q-r}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}{\displaystyle \frac{a^p.a^q}{a^r}:a^{2r}}& ={\displaystyle \frac{a^p.a^q}{a^r.a^{2r}}}\\ & ={\displaystyle \frac{a^{p+q}}{a^{r+2r}}}\\ & ={\displaystyle \frac{a^{p+q}}{a^{3r}}}\\ & =a^{p+q-3r}\end{array}\end{array}$.

$\begin{array}{l}\\ 2.& \text{Jika}x=\sqrt[3]{5+\sqrt[3]{8}},\text{maka nilai}\\ & x^3\text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 3\\ \text{b}.& 4\\ \text{c}.& 5\\ \text{d}.& 6\\ \text{e}.& 7\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}x& =\sqrt[3]{5+\sqrt[3]{8}}\text{dipangkatkan 3}\\ & \text{masing-masing ruas}\\ x^3& ={\left(\sqrt[3]{5+\sqrt[3]{8}}\right)}^3\\ & =5+\sqrt[3]{8}\\ & =5+\sqrt[3]{2^3}=5+2=7\end{array}\end{array}$.

$\begin{array}{l}\\ 3.& \text{Jika}{4}^a\times 4^b=64\text{dan}{\displaystyle \frac{4^a}{4^b}=16}\\ & \text{maka nilai dari}a:b\text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 5\\ \text{b}.& {\displaystyle \frac{5}{4}}\\ \text{c}.& {\displaystyle \frac{1}{3}}\\ \text{d}.& 3\\ \text{e}.& {\displaystyle \frac{3}{4}}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}& \text{Diketahui}\\ & \bullet 4^a\times 4^b=64\iff 4^{a+b}=4^3\iff a+b=3....(1)\\ & \bullet {\displaystyle \frac{4^a}{4^b}=16\iff 4^{a-b}=4^2\iff a-b=2...........(2)}\\ & \text{Dari persamaan}(1)\mathrm{\&}(2)\text{akan didapatkan}\\ & \begin{array}{|ll|}\hline \begin{array}{lll}a+b& =& 3\\ a-b& =& 2& +\\ 2a& =& 5\\ a& =& {\displaystyle \frac{5}{2}}\end{array}& \begin{array}{rll}a+b& =& 3\\ a-b& =& 2& -\\ 2b& =& 1\\ b& =& {\displaystyle \frac{1}{2}}\end{array}\\ \hline\end{array}\\ & \text{Sehingga nilai}a:b={\displaystyle \frac{a}{b}={\displaystyle \frac{{\displaystyle \frac{5}{2}}}{{\displaystyle \frac{1}{2}}}=5}}\end{array}\end{array}$.

$\begin{array}{l}\\ 4.& (\mathbf{\text{SPMB 2003}})\\ & \text{Jika}a\ne 0,\text{maka nilai}{\displaystyle \frac{(-2a{)}^3(2a{)}^{-\frac{2}{3}}}{{\left(16a^4\right)}^{\frac{1}{3}}}\text{adalah ... .}}\\ & \begin{array}{l}\\ \text{a}.& -2^{2}a\\ \text{b}.& {\displaystyle -2a}\\ \text{c}.& {\displaystyle 2a^2}\\ \text{d}.& 2^{2}a\\ \text{e}.& {\displaystyle -2a^2}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}{\displaystyle \frac{(-2a{)}^3(2a{)}^{-\frac{2}{3}}}{{\left(16a^4\right)}^{\frac{1}{3}}}}& ={\displaystyle \frac{-8a^3}{(2a{)}^{\frac{2}{3}}.{\left(16a^4\right)}^{\frac{1}{3}}}}\\ & =-{\displaystyle \frac{(2a{)}^3}{(2a{)}^{\frac{2}{3}}.(2a{)}^{\frac{4}{3}}}}\\ & =-{\displaystyle \frac{(2a{)}^3}{(2a{)}^{\frac{2}{3}+\frac{4}{3}}}}\\ & =-{\displaystyle \frac{(2a{)}^3}{(2a{)}^2}}\\ & =-(2a{)}^{3-2}\\ & =-2a\end{array}\end{array}$.

$\begin{array}{l}\\ 5.& \text{Bentuk sederhana dari}{\displaystyle \frac{4^{n+3}-4^{n+1}}{4(4^{n-1})}}\\ & \text{adalah ... .}\\ & \begin{array}{l}\\ \text{a}.& 64\\ \text{b}.& 60\\ \text{c}.& 18\\ \text{d}.& 16\\ \text{e}.& 15\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}{\displaystyle \frac{4^{n+3}-4^{n+1}}{4(4^{n-1})}}& ={\displaystyle \frac{4^n{.4}^3-4^n.4}{4.{\displaystyle \frac{4^n}{4}}}}\\ & ={\displaystyle \frac{4^n(64-4)}{4^n}=60}\end{array}\end{array}$.