To convert the expression [tex]\(\left(32 a^{10} b^{\frac{5}{2}}\right)^{\frac{2}{5}}\)[/tex] into its radical form, we need to understand the relationship between exponents and radicals. The general rule is that an expression of the form [tex]\(x^{\frac{m}{n}}\)[/tex] can be rewritten as a radical [tex]\(\sqrt[n]{x^m}\)[/tex].
Given the expression:
[tex]\[ \left(32 a^{10} b^{\frac{5}{2}}\right)^{\frac{2}{5}} \][/tex]
Let's identify the values of [tex]\(m\)[/tex] and [tex]\(n\)[/tex] in this expression:
- [tex]\(x = 32 a^{10} b^{\frac{5}{2}}\)[/tex]
- [tex]\(m = 2\)[/tex]
- [tex]\(n = 5\)[/tex]
Applying the rule [tex]\(x^{\frac{m}{n}} = \sqrt[n]{x^m}\)[/tex], we can rewrite the expression as:
[tex]\[ \sqrt[5]{\left(32 a^{10} b^{\frac{5}{2}}\right)^2} \][/tex]
Now let's match this with the given answer choices:
A. [tex]\(\sqrt[5]{\left(32 a^{10} b^{\frac{5}{2}}\right)^2}\)[/tex]
B. [tex]\(\sqrt{\left(32 a^{10} b^{\frac{5}{2}}\right)^5}\)[/tex]
C. [tex]\(\sqrt{\left(32 a^{10} b^{\frac{5}{2}}\right)^{\frac{2}{5}}}\)[/tex]
D. [tex]\(\sqrt{\left(32 a^{10} b^{\frac{5}{2}}\right)^{\frac{1}{5}}}\)[/tex]
The correct answer is clearly:
A. [tex]\(\sqrt[5]{\left(32 a^{10} b^{\frac{5}{2}}\right)^2}\)[/tex]
