To simplify the expression \((-3 a^{1/4})(9 a)^{3/2}\), follow these steps:

### Step 1: Simplify each part of the expression separately.

1. **Simplify \((9 a)^{3/2}\)**:
   \[
   (9 a)^{3/2} = 9^{3/2} \cdot a^{3/2}
   \]
   - \(9^{3/2}\) can be rewritten as \((9^{1/2})^3\). Since \(9^{1/2} = 3\), this becomes:
   \[
   9^{3/2} = 3^3 = 27
   \]
   So, \((9 a)^{3/2} = 27 a^{3/2}\).

### Step 2: Multiply the simplified terms together.

Now multiply the terms \(-3 a^{1/4}\) and \(27 a^{3/2}\):

\[
(-3 a^{1/4})(27 a^{3/2}) = -3 \cdot 27 \cdot a^{1/4} \cdot a^{3/2}
\]

### Step 3: Simplify the multiplication.

1. Multiply the constants:
   \[
   -3 \times 27 = -81
   \]

2. Combine the powers of \(a\):
   \[
   a^{1/4} \cdot a^{3/2} = a^{1/4 + 3/2}
   \]
   Convert \(3/2\) to quarters: \(3/2 = 6/4\), so:
   \[
   a^{1/4 + 6/4} = a^{7/4}
   \]

### Final Answer:
\[
(-3 a^{1/4})(9 a)^{3/2} = -81 a^{7/4}
\]

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you simplify expressions with fractional exponents?
2. What are the properties of exponents that are most useful in simplifying expressions?
3. How would this expression change if the exponent was negative?
4. What is the geometric interpretation of fractional exponents?
5. How can you express \(a^{7/4}\) in radical form?

**Tip:** Always break down complex expressions by simplifying each part individually before combining them.

Learn how to simplify the expression (-3 a^(1/4))(9 a)^(3/2) step-by-step. This problem involves exponents and fractional exponents. Suitable for high school students studying algebra.

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