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The problem asks to find the coefficient of \(x^4 y^{-5}\) from the expansion of \(\left( 2x^2 - \frac{1}{2y} \right)^7\).

We can use the **binomial theorem** to expand this expression, which states:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

Here, \(a = 2x^2\) and \(b = -\frac{1}{2y}\), and \(n = 7\). Using the binomial expansion, we want to find the term where the powers of \(x\) and \(y\) are \(x^4 y^{-5}\).

To do that, the general term in the expansion will be:

\[
\binom{7}{k} (2x^2)^{7-k} \left(-\frac{1}{2y}\right)^k
\]

Now, let's calculate the exponents of \(x\) and \(y\) for each term:

- The \(x\)-term from \((2x^2)^{7-k}\) is \(x^{2(7-k)} = x^{14 - 2k}\).
- The \(y\)-term from \(\left(-\frac{1}{2y}\right)^k\) is \(y^{-k}\).

We need to find the values of \(k\) that satisfy the conditions:
- \(14 - 2k = 4\) (for \(x^4\))
- \(-k = -5\) (for \(y^{-5}\))

From \(-k = -5\), we get \(k = 5\).

Now, substitute \(k = 5\) into \(14 - 2k = 4\), which also holds true, confirming that \(k = 5\) is correct.

Finally, the coefficient of the term corresponding to \(x^4 y^{-5}\) is:

\[
\binom{7}{5} (2x^2)^2 \left(-\frac{1}{2y}\right)^5
\]

This simplifies to:

\[
\binom{7}{5} \cdot 4 \cdot \left(-\frac{1}{2}\right)^5
\]

Now let's calculate this coefficient.The coefficient of the term \(x^4 y^{-5}\) in the expansion of \(\left( 2x^2 - \frac{1}{2y} \right)^7\) is \(-2.625\).

Do you need further clarification or details on any part of this solution?

Here are 5 related questions to explore this topic further:
1. How is the binomial theorem applied for non-integer exponents?
2. What is the significance of the binomial coefficient \(\binom{n}{k}\) in expansions?
3. How would the expansion change if the power of \(y\) was different in the expression?
4. Can the binomial theorem be used for more than two terms?
5. What are the typical uses of binomial expansion in algebra and calculus?

**Tip:** When solving binomial expansion problems, always break the problem down into simpler parts, identifying the powers of variables before dealing with the coefficients.

Tentukan koefisien x^4 y^-5 dari (2x^2 - 1/(2y))^7.

Learn how to find the coefficient of x^4 y^-5 in the binomial expansion of (2x^2 - 1/(2y))^7 using the binomial theorem. This problem explores exponents and binomial expansion, suitable for students in Grades 10-12.

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