Select the correct answer.

Which of the following is the correct radical form of this expression?
[tex]\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}[/tex]

A. [tex]\left(\sqrt[6]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^5[/tex]
B. [tex]\left(\sqrt[5]{\frac{p^{12} q^{\frac{3}{2}}}{64}}\right)^6[/tex]
C. [tex]\sqrt{\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{5}{6}}}[/tex]
D. [tex]\sqrt{\left(\frac{p^{12} q^{\frac{3}{2}}}{64}\right)^{\frac{1}{6}}}[/tex]



Answer :

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]