To solve the expression \((27a^{12})^{-\frac{4}{3}}\), we can follow these steps:

### Step 1: Simplify the expression inside the parentheses.
\[ 27a^{12} \]

### Step 2: Apply the exponent \(-\frac{4}{3}\) to both 27 and \(a^{12}\).

Using the property \((x^m)^n = x^{m \cdot n}\):

- For the constant 27, we can rewrite it as \(3^3\). 
\[ 27 = 3^3 \]
- Apply the exponent \(-\frac{4}{3}\) to \(3^3\):
\[ (3^3)^{-\frac{4}{3}} = 3^{3 \cdot -\frac{4}{3}} = 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \]

- For the variable \(a^{12}\), apply the exponent \(-\frac{4}{3}\):
\[ (a^{12})^{-\frac{4}{3}} = a^{12 \cdot -\frac{4}{3}} = a^{-16} \]

### Step 3: Combine the results.
The expression becomes:
\[ \frac{1}{81} \cdot a^{-16} \]

### Step 4: Rewrite the expression using positive exponents.
\[ \frac{1}{81a^{16}} \]

So, the simplified expression is:
\[
\frac{1}{81a^{16}}
\]

Would you like any additional details on the steps, or do you have any questions?

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Here are 8 follow-up questions:

1. How would you simplify \((8b^6)^{\frac{2}{3}}\)?
2. Can you solve \((x^2y^3)^{-\frac{1}{2}}\)?
3. How does the negative exponent affect the outcome of the expression?
4. What is the general rule for applying fractional exponents to an expression?
5. How do you handle expressions with multiple variables raised to a fractional exponent?
6. Can you simplify \((64c^{18})^{-\frac{1}{2}}\)?
7. What happens when the exponent is positive in a similar expression, e.g., \((16d^4)^{\frac{3}{4}}\)?
8. How would you approach \((125e^{10})^{\frac{1}{3}}\)?

**Tip:** Always rewrite constants as powers of smaller bases when possible; it simplifies the process of applying fractional exponents.

Learn how to simplify the expression (27a^12)^(-4/3) with detailed step-by-step instructions. This problem covers exponentiation and fractional exponents, illustrating how to apply these concepts effectively. Suitable for Grades 7-9 students and anyone looking to understand fractional exponents.

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