To simplify \((\frac{y}{2} - 4z)^3\), we can expand the cube using the binomial expansion formula. The binomial expansion for \((a - b)^3\) is given by:

\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\]

Here, \(a = \frac{y}{2}\) and \(b = 4z\). Applying this to the binomial expansion:

\[
\left(\frac{y}{2} - 4z\right)^3 = \left(\frac{y}{2}\right)^3 - 3\left(\frac{y}{2}\right)^2(4z) + 3\left(\frac{y}{2}\right)(4z)^2 - (4z)^3
\]

Now, simplify each term:

1. \(\left(\frac{y}{2}\right)^3 = \frac{y^3}{8}\)
2. \(-3\left(\frac{y}{2}\right)^2(4z) = -3 \cdot \frac{y^2}{4} \cdot 4z = -3y^2z\)
3. \(3\left(\frac{y}{2}\right)(4z)^2 = 3 \cdot \frac{y}{2} \cdot 16z^2 = 24yz^2\)
4. \(-(4z)^3 = -64z^3\)

So the expanded expression is:

\[
\frac{y^3}{8} - 3y^2z + 24yz^2 - 64z^3
\]

This is the fully expanded form of \((\frac{y}{2} - 4z)^3\).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How do you apply the binomial theorem for higher powers like \( (a - b)^n \)?
2. What happens when you cube a binomial with more complex terms?
3. Can you factor the result of \((\frac{y}{2} - 4z)^3\) back to the original form?
4. How would you apply the binomial expansion if there was a constant in front of the binomial?
5. What is the difference between expanding and simplifying an expression?

**Tip:** When expanding binomials, keeping track of negative signs carefully is crucial to avoid mistakes.

Expand and simplify the expression (y/2 - 4z)³ using the binomial theorem. This solution provides a detailed breakdown of key algebraic concepts, including binomial expansion and polynomial simplification, suitable for students in Grades 9-11.

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