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Moishe Lettvin · Answer 1 · 2023-01-18T23:43:15+0000

Answer::

$\sqrt[30]{r^(29)}$

Explanation:

Given the expression:

$\sqrt[5]{r^4}\sqrt[6]{r}$

First, rewrite the expression using the fractional index law:

$\begin{gathered} \sqrt[n]{x}=x^{(1)/(n)} \\ \implies\sqrt[5]{r^4}=r^{(4)/(5)};\text{ and} \\ \sqrt[6]{r}=r^{(1)/(6)} \end{gathered}$

Therefore:

$\sqrt[5]{r^4}*\sqrt[6]{r}=r^{(4)/(5)}* r^{(1)/(6)}$

Use the multiplication law of exponents:

$\begin{gathered} a^x* a^y=a^(x+y) \\ \implies r^{(4)/(5)}* r^{(1)/(6)}=r^{(4)/(5)+(1)/(6)} \\ (4)/(5)+(1)/(6)=(24+5)/(30)=(29)/(30) \\ \operatorname{\implies}r^{(4)/(5)}* r^{(1)/(6)}=r^{(4)/(5)+(1)/(6)}=r^{(29)/(30)} \end{gathered}$

The resulting expression can be rewrittem further:

$\begin{gathered} r^{(29)/(30)}=(r^(29))^{(1)/(30)} \\ =\sqrt[30]{r^(29)} \end{gathered}$

The single radical is:

$\sqrt[30]{r^(29)}$

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