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To solve the problem of finding the coefficient of \(x^{-6}\) in the expansion of \( \left( \frac{4x}{5} + \frac{5}{2x^2} \right)^9 \), we can use the binomial expansion formula.

### Binomial Expansion Formula:
The binomial expansion of \( (a + b)^n \) is given by:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case, we have \(a = \frac{4x}{5}\), \(b = \frac{5}{2x^2}\), and \(n = 9\).

We need to expand this binomial and identify the term that contains \(x^{-6}\).

### Step 1: General Term in the Expansion
The general term in the expansion is given by:
\[
T_k = \binom{9}{k} \left( \frac{4x}{5} \right)^{9-k} \left( \frac{5}{2x^2} \right)^k
\]

### Step 2: Simplifying the General Term
Simplifying the general term, we get:
\[
T_k = \binom{9}{k} \cdot \left( \frac{4^{9-k} x^{9-k}}{5^{9-k}} \right) \cdot \left( \frac{5^k}{2^k x^{2k}} \right)
\]
\[
T_k = \binom{9}{k} \cdot \frac{4^{9-k} \cdot 5^k}{5^{9-k} \cdot 2^k} \cdot \frac{x^{9-k}}{x^{2k}}
\]
\[
T_k = \binom{9}{k} \cdot \frac{4^{9-k} \cdot 5^k}{5^{9-k} \cdot 2^k} \cdot x^{9 - 3k}
\]

### Step 3: Finding \(k\) for \(x^{-6}\)
We need to find the value of \(k\) such that the power of \(x\) is \(-6\). From the general term, the power of \(x\) is \(9 - 3k\), so we set this equal to \(-6\):
\[
9 - 3k = -6
\]
\[
3k = 15
\]
\[
k = 5
\]

### Step 4: Substituting \(k = 5\) into the General Term
Now, substitute \(k = 5\) into the general term:
\[
T_5 = \binom{9}{5} \cdot \frac{4^{9-5} \cdot 5^5}{5^{9-5} \cdot 2^5} \cdot x^{9 - 3(5)}
\]
\[
T_5 = \binom{9}{5} \cdot \frac{4^4 \cdot 5^5}{5^4 \cdot 2^5} \cdot x^{-6}
\]
\[
T_5 = \binom{9}{5} \cdot \frac{4^4}{2^5} \cdot x^{-6}
\]

### Step 5: Calculating the Coefficient
First, calculate the binomial coefficient:
\[
\binom{9}{5} = \frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1} = 126
\]

Next, calculate \(4^4 = 256\) and \(2^5 = 32\), so the coefficient becomes:
\[
T_5 = 126 \cdot \frac{256}{32} = 126 \cdot 8 = 1008
\]

### Final Answer:
The coefficient of \(x^{-6}\) is \( \boxed{1008} \).

Do you have any questions or would you like more details on any part of the solution?

Here are 5 related questions you can explore:
1. How does the binomial expansion work for negative exponents?
2. How can we use the binomial theorem to solve problems involving fractional exponents?
3. How does the binomial coefficient change when increasing the power of the binomial expansion?
4. What happens to the expansion if the variables in the terms have different powers?
5. Can the binomial expansion be applied to trinomials or polynomials with more than two terms?

**Tip:** The binomial expansion formula is particularly useful when dealing with expressions raised to large powers, as it helps break them down into manageable terms.

The coefficient of x^{-6} in the expansion of (4x/5 + 5/2x^2)^9 is ______.

Learn how to determine the coefficient of x^{-6} in the binomial expansion of (4x/5 + 5/2x^2)^9. This solution uses the binomial theorem, exponent rules, and step-by-step simplifications. Suitable for advanced high school or early college students.

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