Sure, let's break down the expression [tex]\((\sqrt{32})(\sqrt[5]{64})\)[/tex] step by step.
1. Convert the square root and fifth root into exponent form:
- [tex]\(\sqrt{32}\)[/tex] is the same as [tex]\(32^{1/2}\)[/tex].
- [tex]\(\sqrt[5]{64}\)[/tex] is the same as [tex]\(64^{1/5}\)[/tex].
2. Express 32 and 64 in terms of powers of 2:
- [tex]\(32 = 2^5\)[/tex],
- [tex]\(64 = 2^6\)[/tex].
3. Rewrite the expression using these bases:
[tex]\[
(32^{1/2})(64^{1/5}) = (2^5)^{1/2} \cdot (2^6)^{1/5}.
\][/tex]
4. Apply the power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(2^5)^{1/2} = 2^{5/2},
\][/tex]
[tex]\[
(2^6)^{1/5} = 2^{6/5}.
\][/tex]
5. Multiply these expressions:
[tex]\[
2^{5/2} \cdot 2^{6/5}.
\][/tex]
6. When multiplying with the same base, you add the exponents:
[tex]\[
2^{5/2 + 6/5}.
\][/tex]
7. Find a common denominator to add these exponents:
[tex]\[
\frac{5}{2} + \frac{6}{5} = \frac{25}{10} + \frac{12}{10} = \frac{37}{10}.
\][/tex]
8. Therefore, the expression simplifies to:
[tex]\[
2^{37/10}.
\][/tex]
9. To recognize which option this matches, recall that [tex]\(2^{37/10}\)[/tex] can be rewritten to fit one of the forms given in the choices.
Observe that [tex]\(2^{37/10}\)[/tex] can be expressed in the form [tex]\(8 \cdot 2^{7/10}\)[/tex]:
[tex]\[
2^{37/10} = 2^{3 + 7/10} = 2^3 \cdot 2^{7/10} = 8 \cdot 2^{7/10}.
\][/tex]
Therefore, the equivalent expression matches option D:
[tex]\[8\left(\sqrt[10]{2^7}\right).\][/tex]
Thus, the correct answer is [tex]\( \boxed{D} \)[/tex].
