Answer::
[tex]\sqrt[30]{r^{29}}[/tex]
Explanation:
Given the expression:
[tex]\sqrt[5]{r^4}\sqrt[6]{r}[/tex]
First, rewrite the expression using the fractional index law:
[tex]\begin{gathered} \sqrt[n]{x}=x^{\frac{1}{n}} \\ \implies\sqrt[5]{r^4}=r^{\frac{4}{5}};\text{ and} \\ \sqrt[6]{r}=r^{\frac{1}{6}} \end{gathered}[/tex]
Therefore:
[tex]\sqrt[5]{r^4}\times\sqrt[6]{r}=r^{\frac{4}{5}}\times r^{\frac{1}{6}}[/tex]
Use the multiplication law of exponents:
[tex]\begin{gathered} a^x\times a^y=a^{x+y} \\ \implies r^{\frac{4}{5}}\times r^{\frac{1}{6}}=r^{\frac{4}{5}+\frac{1}{6}} \\ \frac{4}{5}+\frac{1}{6}=\frac{24+5}{30}=\frac{29}{30} \\ \operatorname{\implies}r^{\frac{4}{5}}\times r^{\frac{1}{6}}=r^{\frac{4}{5}+\frac{1}{6}}=r^{\frac{29}{30}} \end{gathered}[/tex]
The resulting expression can be rewrittem further:
[tex]\begin{gathered} r^{\frac{29}{30}}=(r^{29})^{\frac{1}{30}} \\ =\sqrt[30]{r^{29}} \end{gathered}[/tex]
The single radical is:
[tex]\sqrt[30]{r^{29}}[/tex]
