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When rewriting a radical into exponential form, the exponent of the radicand is the numerator of the rational exponent, and the index of the radical is the denominator of the rational exponent. $\begin{gathered} \sqrt[{\color{#009600}{n}}]{a}=a^\frac{1}{{\color{#009600}{n}}} \quad \text{ and } \quad \sqrt[{\color{#009600}{n}}]{a^{\color{#FF0000}{m}}}=a^\frac{{\color{#FF0000}{m}}}{{\color{#009600}{n}}} \end{gathered}$ With this in mind, we can rewrite the given expression. $\begin{gathered} &\sqrt[{\color{#009600}{4}}]{\left(7p^6q^9\right)^{\color{#FF0000}{1}}} \quad \Leftrightarrow \quad \left(7p^6q^9\right)^\frac{{\color{#FF0000}{1}}}{{\color{#009600}{4}}} \end{gathered}$ Now, let's simplify this expression as much as possible.
$\left(7p^6q^9\right)^\frac{1}{4}$
$7^\frac{1}{4}\left(p^6\right)^{ \frac{1}{4}} \left(q^9\right)^{\frac{1}{4}}$
$7^\frac{1}{4}p^{6\cdot \frac{1}{4}} q^{9\cdot \frac{1}{4}}$
$7^\frac{1}{4}p^\frac{6}{4} q^ \frac{9}{4}$
$7^\frac{1}{4}p^\frac{3}{2}q^\frac{9}{4}$