We are tasked with simplifying the expression:

\[
\left( \frac{14^7}{9} \right)^{-\frac{4}{6}} = \left( \frac{14^7}{9} \right)^{-\frac{2}{3}}
\]

### Step-by-step solution:

1. **Simplifying the exponent**: The exponent \(\frac{4}{6}\) simplifies to \(\frac{2}{3}\).

   \[
   \left( \frac{14^7}{9} \right)^{-\frac{2}{3}} = \frac{1}{\left( \frac{14^7}{9} \right)^{\frac{2}{3}}}
   \]

2. **Using properties of exponents**: Apply the exponent to both the numerator and denominator:

   \[
   \frac{1}{\left( \frac{14^7}{9} \right)^{\frac{2}{3}}} = \frac{1}{\frac{(14^7)^{\frac{2}{3}}}{9^{\frac{2}{3}}}}
   \]

   This becomes:

   \[
   \frac{9^{\frac{2}{3}}}{(14^7)^{\frac{2}{3}}}
   \]

3. **Simplifying powers**: Simplify each term:

   \[
   (14^7)^{\frac{2}{3}} = 14^{\frac{14}{3}}, \quad 9^{\frac{2}{3}} = \left( 3^2 \right)^{\frac{2}{3}} = 3^{\frac{4}{3}}
   \]

4. **Final form**: The simplified expression is:

   \[
   \frac{3^{\frac{4}{3}}}{14^{\frac{14}{3}}}
   \]

This is the simplest form of the expression.

Would you like a more detailed explanation of any step?

### 5 Related Questions:
1. How can we simplify expressions with negative exponents further?
2. What are the properties of fractional exponents and how do they apply to roots?
3. How can we simplify \((a^m)^{n/p}\) for any values of \(a, m, n,\) and \(p\)?
4. What is the relationship between exponents and radicals in simplifying expressions?
5. How do fractional exponents affect numerators and denominators?

### Tip:


Understand how to simplify the expression (14^7/9)^-4/6 with a step-by-step solution using fractional exponents and negative exponents. This problem demonstrates exponentiation laws and is suitable for students in Grades 9-11.

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