Exercise 1.4.1 (1-2)

Question

\begin{enumerate}
  \item \begin{enumerate}[label=(\alph*)]
\item $\frac{-2^{6}}{4^{3}}$

\item  $\frac{(-3)^{6}}{9^{6}}$

\item  $\frac{1}{\sqrt[4]{x^{5}}}$

\item  $\frac{x^{3} \cdot x^{n}}{x^{n+1}}$

\item  $b^{3}\left(3 b^{-1}\right)^{-2}$

\item  $\frac{2 x^{2} y}{\left(3 x^{-2} y\right)^{2}}$
\end{enumerate}

\item \begin{enumerate}[label=(\alph*)]
\item $\frac{\sqrt[3]{4}}{\sqrt[3]{108}}$

\item  $27^{2 / 3}$

\item  $2 x^{2}\left(3 x^{5}\right)^{2}$

\item  $\left(2 x^{-2}\right)^{-3} x^{-3}$

\item  $\frac{3 a^{3 / 2} \cdot a^{1 / 2}}{a^{-1}}$

\item  $\frac{\sqrt{a \sqrt{b}}}{\sqrt[3]{a b}}$
\end{enumerate}
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}\item \begin{enumerate}
  \item [(a)] $$\frac{-2^{6}}{4^{3}}=\frac{-2^{6}}{\left(2^{2}\right)^{3}}=\frac{-2^{6}}{2^{6}}=-1.$$
  \item [(b)] $$\frac{(-3)^{6}}{9^{6}}=\frac{3^{6}}{9^{6}}=\frac{3^{6}}{\left(3^{2}\right)^{6}}=\frac{3^{6}}{\left(3^{6}\right)^{2}}=\frac{1}{3^{6}}.$$
  \item [(c)] $$\frac{1}{\sqrt[4]{x^{5}}}=\frac{1}{x^{\frac{5}{4}}}=\frac{1}{x \cdot x^{\frac{1}{4}}}=\frac{1}{x \sqrt[4]{x}}.$$
  \item [(d)] $$\frac{x^{3} \cdot x^{n}}{x^{n+1}}=\frac{x^{3}}{x}=x^{2}.$$
  \item [(e)] $$b^{3}\left(3 b^{-1}\right)^{-2}=b^{3} \cdot \frac{1}{9} \cdot b^{2}=\frac{1}{9} b^{5}.$$
  \item [(f)] $$\frac{2 x^{2} y}{\left(3 x^{-2} y\right)^{2}}=\frac{2 x^{2} y}{9 x^{-4} y^{2}}=\frac{2}{9} x^{6} y^{-1}.$$
\end{enumerate}
\item \begin{enumerate}[label=(\alph*)]
\item $$\frac{\sqrt[3]{4}}{\sqrt[3]{108}}=\sqrt[3]{\frac{4}{108}}=\sqrt[3]{\frac{1}{27}}=\frac{1}{3}.$$
  \item $$27^{2 / 3}=\sqrt[3]{27^{2}}=\sqrt[3]{\left(3^{3}\right)^{2}}=\sqrt[3]{\left(3^{2}\right)^{3}}=3^{2}=9.$$
  \item $$2 x^{2}\left(3 x^{5}\right)^{2}=2 x^{2} \cdot 9 x^{10}=18 x^{12}.$$
  \item $$\left(2 x^{-2}\right)^{-3} x^{-3}=\left(2 x^{-2} \cdot x\right)^{-3}=\left(2 x^{-1}\right)^{-3}=\frac{1}{8} x^{3}.$$
  \item $$\frac{3 a^{3 / 2} \cdot a^{1 / 2}}{a^{-1}}=\frac{3 a^{(3 / 2+1 / 2)}}{a^{-1}}=\frac{3 a^{2}}{a^{-1}}=3 a^{2-(-1)}=3 a^{3}$$
  \item $$\frac{\sqrt{a\sqrt{b}}}{\sqrt[3]{ab}}=\frac{\sqrt{a}\cdot \sqrt[4]{b}}{\sqrt[3]{a}\cdot \sqrt[3]{b}}=\frac{\sqrt{a}}{\sqrt[3]{a}}\cdot \frac{\sqrt[4]{b}}{\sqrt[3]{b}}=\frac{a^{\frac{1}{2}}}{a^{\frac{1}{3}}}\cdot \frac{b^{\frac{1}{4}}}{b^{\frac{1}{3}}}=a^{1/2-1/3}\cdot b^{1/4-1/3}=a^{1/6}\cdot b^{-1/12}=\frac{\sqrt[6]{x}}{\sqrt[12]{x}}.$$
\end{enumerate}
\end{enumerate}

Exercise 1.4.1 (1-2)

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Exercise 1.4.1 (1-2)

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