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We are tasked with solving the following mathematical expression:

\[
(\sqrt{32})(\sqrt[5]{64})
\]

To begin, let's simplify both components of the product:

1. **Simplify \(\sqrt{32}\):**
   \[
   \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}
   \]

2. **Simplify \(\sqrt[5]{64}\):**
   \[
   \sqrt[5]{64} = \sqrt[5]{2^6} = 2^{6/5}
   \]

Now, we compute the product of these two simplified expressions:

\[
(4\sqrt{2}) \times (2^{6/5}) = 4 \times 2^{1/2} \times 2^{6/5}
\]

Using the property of exponents, \(2^{a} \times 2^{b} = 2^{a+b}\):

\[
4 \times 2^{1/2 + 6/5} = 4 \times 2^{(5/10 + 12/10)} = 4 \times 2^{17/10}
\]

Next, express 4 as \(2^2\):

\[
2^2 \times 2^{17/10} = 2^{2 + 17/10} = 2^{(20/10 + 17/10)} = 2^{37/10}
\]

This matches with the answer choices. Since the form of the expression is \( 8 \times 2^{k} \) where \(k = 7/5\), this corresponds to **option D**:

\[
\boxed{8 \times 2^{10/27}}.
\]

Would you like further details or have any questions? Here are 5 related questions for practice:

1. How do you simplify radical expressions involving prime factorization?
2. What are the properties of exponents used to combine roots and powers?
3. How do you simplify fractional exponents like \(2^{6/5}\)?
4. What is the significance of multiplying powers of the same base?
5. Can you explain the process of converting between radicals and fractional exponents?

**Tip:** Always simplify radical expressions by first looking for perfect squares or cubes!

Which of the following is equivalent to (√32)(√⁵64)?

Learn how to simplify the expression (√32)(√⁵64) using properties of exponents and radicals. This problem focuses on simplifying expressions involving square and fifth roots, and is suitable for students in Grades 9-12.

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