To determine the correct radical form of the expression [tex]\(\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}\)[/tex], let's analyze the given expression step by step.
Starting with the expression:
[tex]\[
\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}
\][/tex]
The expression involves raising the fraction [tex]\(\frac{p^{12} q^{\frac{3}{2}}}{64}\)[/tex] to the power of [tex]\(\frac{5}{6}\)[/tex]. In order to convert this into a radical form, we utilize the property that the exponent [tex]\(\frac{a}{b}\)[/tex] can be interpreted as taking the [tex]\(b\)[/tex]-th root and then raising to the power [tex]\(a\)[/tex].
Thus, [tex]\(\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}\)[/tex] which means:
[tex]\[
\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}} = \left(\sqrt[6]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^5
\][/tex]
This matches with option A:
[tex]\[
A. \left(\sqrt[6]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^5
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
