To calculate $\frac{z}{w}$ where $z = 72 + 4i$ and $w = 4 - 7i$, we will follow these steps:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator

The conjugate of $w = 4 - 7i$ is $4 + 7i$.

$\frac{z}{w} = \frac{72 + 4i}{4 - 7i} \times \frac{4 + 7i}{4 + 7i}$

Step 2: Simplify the denominator

Using the difference of squares formula:

$(4 - 7i)(4 + 7i) = 4^2 - (-7i)^2 = 16 + 49 = 65$

So, the denominator becomes 65.

Step 3: Multiply the numerators

Now multiply $(72 + 4i)(4 + 7i)$:

$(72 + 4i)(4 + 7i) = 72(4 + 7i) + 4i(4 + 7i)$

First, distribute the terms:

$= 72 \times 4 + 72 \times 7i + 4i \times 4 + 4i \times 7i$ $= 288 + 504i + 16i + 28i^2$

Since $i^2 = -1$, substitute:

$= 288 + 504i + 16i + 28(-1)$ $= 288 + 504i + 16i - 28$ $= 260 + 520i$

Step 4: Final division

Now divide by 65:

$\frac{260 + 520i}{65} = \frac{260}{65} + \frac{520i}{65}$ $= 4 + 8i$

Thus, $\frac{z}{w} = 4 + 8i$.

Final Answer:

$\frac{z}{w} = 4 + 8i$

Would you like further details on any steps?

---

Here are 5 related questions to explore further:

1. What is the magnitude of the complex number $4 + 8i$?
2. How do you divide complex numbers using polar form?
3. What is the geometric interpretation of multiplying by a complex conjugate?
4. How do you convert a complex number from rectangular form to polar form?
5. How does multiplying two complex numbers affect their angles and magnitudes?

Tip: When multiplying complex numbers, use their polar forms to simplify the process by focusing on the magnitudes and angles.