if z = 5-2i  and w=3+5i  then find the quotient of z/w

To find the quotient of the complex numbers \( z = 5 - 2i \) and \( w = 3 + 5i \), we divide \( z \) by \( w \). This process involves multiplying both the numerator and the denominator by the conjugate of the denominator.

### Step 1: Write the quotient
\[
\frac{z}{w} = \frac{5 - 2i}{3 + 5i}
\]

### Step 2: Multiply the numerator and denominator by the conjugate of the denominator
The conjugate of \( w = 3 + 5i \) is \( 3 - 5i \).

\[
\frac{z}{w} = \frac{(5 - 2i) \times (3 - 5i)}{(3 + 5i) \times (3 - 5i)}
\]

### Step 3: Expand the numerator and denominator

**Numerator:**
\[
(5 - 2i)(3 - 5i) = 5 \times 3 + 5 \times (-5i) - 2i \times 3 - 2i \times (-5i)
\]
\[
= 15 - 25i - 6i + 10i^2
\]
Since \( i^2 = -1 \),
\[
= 15 - 25i - 6i - 10 = 5 - 31i
\]

**Denominator:**
\[
(3 + 5i)(3 - 5i) = 3^2 - (5i)^2
\]
\[
= 9 - 25i^2
\]
Since \( i^2 = -1 \),
\[
= 9 + 25 = 34
\]

### Step 4: Write the quotient in standard form
\[
\frac{z}{w} = \frac{5 - 31i}{34} = \frac{5}{34} - \frac{31}{34}i
\]

So, the quotient is:

\[
\frac{z}{w} = \frac{5}{34} - \frac{31}{34}i
\]

Would you like more details or have any questions?

Here are five related questions you might be interested in:
1. How do you find the modulus of a complex number?
2. What is the geometric interpretation of multiplying two complex numbers?
3. How can you express a complex number in polar form?
4. How do you find the argument of a complex number?
5. What is the significance of the conjugate in complex number division?

**Tip:** When dividing complex numbers, always multiply by the conjugate to simplify the expression.

Learn how to find the quotient of complex numbers z = 5 - 2i and w = 3 + 5i using the method of multiplying by the conjugate. This problem covers the division of complex numbers and the importance of using conjugates in simplifying the expression. Suitable for advanced high school students.

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