$\begin{array}{ll}\\ 1.&\textrm{Bentuk sederhana dari}\: \: \displaystyle \frac{a^{p}.a^{q}}{a^{r}}:a^{2r} \: \: \textrm{adalah ... .}\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&a^{p+q-3r}\\ \textrm{b}.&a^{p+3r-q}\\ \textrm{c}.&a^{p-2q+r}\\ \textrm{d}.&a^{p+q+r}\\ \textrm{e}.&a^{p-3q-r} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}\displaystyle \frac{a^{p}.a^{q}}{a^{r}}:a^{2r}&=\displaystyle \frac{a^{p}.a^{q}}{a^{r}.a^{2r}}\\ &=\displaystyle \frac{a^{p+q}}{a^{r+2r}}\\ &=\displaystyle \frac{a^{p+q}}{a^{3r}}\\ &=\color{red}a^{p+q-3r} \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 2.&\textrm{Jika}\: \: x=\sqrt[3]{5+\sqrt[3]{8}}\: ,\: \: \textrm{maka nilai}\\ &x^{3}\: \textrm{adalah ... .}\\ &\begin{array}{llll}\\ \textrm{a}.&3\\ \textrm{b}.&4\\ \textrm{c}.&5\\ \textrm{d}.&6\\ \color{red}\textrm{e}.&7 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}x&=\sqrt[3]{5+\sqrt[3]{8}}\quad \color{black}\textrm{dipangkatkan 3}\\ &\qquad \color{black}\textrm{masing-masing ruas}\\ x^{3}&=\left ( \sqrt[3]{5+\sqrt[3]{8}} \right )^{3}\\ &=5+\sqrt[3]{8}\\ &=5+\sqrt[3]{2^{3}}=5+2=\color{red}7 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 3.&\textrm{Jika}\: \: 4^{a}\times 4^{b}=64\: \: \textrm{dan}\: \: \displaystyle \frac{4^{a}}{4^{b}}=16\\ &\textrm{maka nilai dari}\: \: a:b\: \: \textrm{adalah ... .}\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&5\\ \textrm{b}.&\displaystyle \frac{5}{4}\\ \textrm{c}.&\displaystyle \frac{1}{3}\\ \textrm{d}.&3\\ \textrm{e}.&\displaystyle \frac{3}{4} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}&\textrm{Diketahui}\\ &\bullet \quad 4^{a}\times 4^{b}=64\color{black}\Leftrightarrow 4^{a+b}=4^{3}\Leftrightarrow a+b=3\: \: ....(1)\\ &\bullet \quad \displaystyle \frac{4^{a}}{4^{b}}=16\color{black}\Leftrightarrow 4^{a-b}=4^{2}\Leftrightarrow a-b=2\: \: ...........(2) \\ &\color{black}\textrm{Dari persamaan}\: \: (1)\: \: \&\: \: (2)\: \: \textrm{akan didapatkan}\\ &\begin{array}{|l|l|}\hline \begin{array}{llllll} a+b&=&3\\ a-b&=&2&+\\\hline 2a&=&5\\ \: \: a&=&\displaystyle \frac{5}{2} \end{array}&\begin{array}{rlllll} a+b&=&3\\ a-b&=&2&-\\\hline 2b&=&1\\ \: \: b&=&\displaystyle \frac{1}{2} \end{array}\\\hline \end{array}\\ &\textrm{Sehingga nilai}\: \: a:b=\displaystyle \frac{a}{b}=\displaystyle \frac{\displaystyle \frac{5}{2}}{\displaystyle \frac{1}{2}}=\color{red}5 \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 4.&(\textbf{SPMB 2003})\\ &\textrm{Jika}\: \: a\neq 0,\: \textrm{maka nilai}\: \: \displaystyle \frac{(-2a)^{3}(2a)^{-\frac{2}{3}}}{\left ( 16a^{4} \right )^{\frac{1}{3}}}\: \: \textrm{adalah ... .}\\ &\begin{array}{llll}\\ \textrm{a}.&-2^{2}a\\ \color{red}\textrm{b}.&\displaystyle -2a\\ \textrm{c}.&\displaystyle 2a^{2}\\ \textrm{d}.&2^{2}a\\ \textrm{e}.&\displaystyle -2a^{2} \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\begin{aligned}\displaystyle \frac{(-2a)^{3}(2a)^{-\frac{2}{3}}}{\left ( 16a^{4} \right )^{\frac{1}{3}}}&=\displaystyle \frac{-8a^{3}}{(2a)^{\frac{2}{3}}.\left ( 16a^{4} \right )^{\frac{1}{3}}}\\ &=-\displaystyle \frac{(2a)^{3}}{(2a)^{\frac{2}{3}}.(2a)^{\frac{4}{3}}}\\ &=-\displaystyle \frac{(2a)^{3}}{(2a)^{\frac{2}{3}+\frac{4}{3}}}\\ &=-\displaystyle \frac{(2a)^{3}}{(2a)^{2}}\\ &=-(2a)^{3-2}\\ &=\color{red}-2a \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 5.&\textrm{Bentuk sederhana dari}\: \: \displaystyle \frac{4^{n+3}-4^{n+1}}{4(4^{n-1})}\: \\ &\textrm{adalah ... .}\\ &\begin{array}{llll}\\ \textrm{a}.&64\\ \color{red}\textrm{b}.&60\\ \textrm{c}.&18\\ \textrm{d}.&16\\ \textrm{e}.&15 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned} \displaystyle \frac{4^{n+3}-4^{n+1}}{4(4^{n-1})}&=\displaystyle \frac{4^{n}.4^{3}-4^{n}.4}{4.\displaystyle \frac{4^{n}}{4}}\\ &=\displaystyle \frac{4^{n}(64-4)}{4^{n}}=\color{red}60\\ \end{aligned}\\ \end{array}$.
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