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?

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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.

Let z = 72 + 4i and w = 4 - 7i calculate z/w

Learn how to divide the complex numbers z = 72 + 4i and w = 4 - 7i with a detailed step-by-step solution. This problem covers key concepts such as using the conjugate, simplifying the denominator, and performing complex number arithmetic. Suitable for students in Grades 10-12.

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