class 9th math 2.3 solution

\[\begin{align}
 (\text{i})\qquad&\sqrt[3]{-64} \\
 & ={{\left( -64 \right)}^{\frac{1}{3}}} \\
  (\text{ii})\qquad&{{2}^{3/5}} \\
 & =\sqrt[5]{{{2}^{3}}} \\
  (\text{iii})\qquad&-{{7}^{1/3}} \\
 & =\sqrt[3]{-7} \\
  (\text{iv})\qquad&{{y}^{-2/3}} \\
 & =\sqrt[3]{{{y}^{-2}}} \\
\end{align}\]

Question#2: Tell whether the following statements are true or false?

\[\begin{align}
   \left( \text{i} \right) \qquad & {{5}^{1/5}}=\sqrt{5}\qquad & \textbf{False} \\
  \left( \text{ii} \right)\qquad & {{2}^{2/3}}=\sqrt[3]{4}\qquad&\textbf{True} \\
  \left( \text{iii} \right) \qquad&\sqrt{49}=\sqrt{7}\qquad&\textbf{False} \\
  \left( \text{iv} \right) \qquad&\sqrt[3]{{{x}^{27}}}={{x}^{3}}\qquad&\textbf{False} \\
\end{align}\]  

\((i)\)

\[\sqrt[3]{-125}\]

Solution:

\[\begin{align}
  & =\sqrt[3]{-125} \\
 & =\sqrt[3]{{{\left( -5 \right)}^{3}}} \\
 & ={{\left( -5 \right)}^{\frac{\cancel{3}}{\cancel{3}}}} \\
 & =-5 \\
\end{align}\]

\((ii)\)

\[\sqrt[4]{32}\]

Solution:

\[\begin{align}
  & =\sqrt[4]{32} \\
 & =\sqrt[4]{{{2}^{4}}\times 2} \\
 & =\sqrt[4]{{{2}^{4}}}\times \sqrt[4]{2} \\
 & ={{2}^{\frac{\cancel{4}}{\cancel{4}}}}\times \sqrt[4]{2} \\
 & =2\times \sqrt[4]{2} \\
 & =2\sqrt[4]{2} \\
\end{align}\]

\((iii)\)

\[\sqrt[5]{\frac{3}{32}}\]

Solution:

\[\begin{align}
  & =\sqrt[5]{\frac{3}{32}} \\
 & =\frac{\sqrt[5]{3}}{\sqrt[5]{32}} \\
 & = \frac{\sqrt[5]{3}}{\sqrt[5]{2^5}}\\  
& =\frac{\sqrt[5]{3}}{{{2}^{\frac{5}{5}}}} \\
 & =\frac{\sqrt[5]{3}}{2} \\
\end{align}\]

\((iv)\)

\[\sqrt[3]{-\frac{8}{27}}\]

Solution:

\[\begin{align}
  & =\sqrt[3]{-\frac{8}{27}} \\
 & =\sqrt[3]{-\frac{{{2}^{3}}}{{{3}^{3}}}} \\
 & =\sqrt[3]{{{\left( -\frac{2}{3} \right)}^{3}}} \\
 & ={{\left( -\frac{2}{3} \right)}^{\frac{3}{3}}} \\
 & =-\frac{2}{3} \\
\end{align}\]