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The given expression is:

\[
(64x)^{1/2} \cdot (-3x^{2/5})
\]

We are asked to simplify it, assuming \( x > 0 \), and to write the final answer without radicals.

### Step 1: Simplify \( (64x)^{1/2} \)
First, break this into two parts:
\[
(64x)^{1/2} = 64^{1/2} \cdot x^{1/2}
\]

Since \( 64^{1/2} = 8 \), this becomes:
\[
(64x)^{1/2} = 8x^{1/2}
\]

### Step 2: Simplify the expression
Now multiply:
\[
8x^{1/2} \cdot (-3x^{2/5})
\]

### Step 3: Combine the powers of \(x\)
Using the laws of exponents, when multiplying terms with the same base, we add the exponents. Thus:
\[
x^{1/2} \cdot x^{2/5} = x^{(1/2 + 2/5)}
\]

We need to add \( \frac{1}{2} \) and \( \frac{2}{5} \). The common denominator is 10:
\[
\frac{1}{2} = \frac{5}{10}, \quad \frac{2}{5} = \frac{4}{10}
\]
\[
\frac{1}{2} + \frac{2}{5} = \frac{5}{10} + \frac{4}{10} = \frac{9}{10}
\]

Thus:
\[
x^{1/2} \cdot x^{2/5} = x^{9/10}
\]

### Step 4: Multiply the constants
Now multiply the constants:
\[
8 \cdot (-3) = -24
\]

### Final Answer:
The simplified expression is:
\[
-24x^{9/10}
\]

Let me know if you'd like further details or explanations!

---

Here are 5 related questions to practice:

1. Simplify \( (36x)^{1/2} \cdot (-2x^{1/3}) \).
2. Simplify \( (81x^2)^{1/2} \cdot (-4x^{1/4}) \).
3. Simplify \( (49y)^{1/3} \cdot (-5y^{2/3}) \).
4. Simplify \( (100x)^{3/2} \cdot (2x^{1/5}) \).
5. Simplify \( (9z^2)^{1/4} \cdot (-7z^{1/2}) \).

**Tip:** Remember that when multiplying powers with the same base, you always add the exponents!

Simplify the following expression. Assume x > 0 and write your answer without radicals: (64x)^(1/2) * (-3x^(2/5))

Learn how to simplify the expression (64x)^(1/2) * (-3x^(2/5)) by applying exponent rules. This problem covers key algebraic concepts, including combining powers with the same base. Suitable for students in Grades 9-12.

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