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For any real number $a,$ the radical expression $\sqrt[n]{a^n}$ can be simplified as follows. $\begin{gathered} \sqrt[{\color{#FF0000}{n}}]{a^{\color{#FF0000}{n}}}= \begin{cases} \phantom{|}a\phantom{|} \text{ if } {\color{#FF0000}{n}} \text{ is odd}\\ |a| \text{ if } {\color{#FF0000}{n}} \text{ is even} \end{cases} \end{gathered}$ Because the given root has an even index and the variables have even exponents, we will use the absolute value symbol to simplify the expression.
$\sqrt[4]{81g^{16}h^{24}}$
$\sqrt[4]{3^4g^{16}h^{24}}$
$\sqrt[4]{3^4g^{4\cdot 4}h^{6\cdot 4}}$
$\sqrt[4]{3^4 \left(g^{4}\right)^4 \left(h^{6}\right)^4}$
$\sqrt[4]{\left(3g^{4}h^{6}\right)^4}$
$\sqrt[n]{a^n}=|a|$
$\left|3g^{4}h^{6}\right|$
Note that $3$ is a positive number. Moreover, note that the expressions $g^4$ and $h^6$ have even exponents. Therefore, they are both positive. Since the product of three positive numbers is positive, the expression $3g^4h^6$ is positive. $\begin{gathered} \left|3g^{4}h^{6}\right| \quad \Leftrightarrow \quad 3g^4h^6 \end{gathered}$