Consider the complex numbers z1=−6−6⋅i and z2=5−6⋅i. Find, giving your answers in the form a+b⋅i, where a,b∈R

:

a)  6⋅z1+3⋅z2=

b)  4⋅z1⋅z2=

c)  z1z2=


Let's solve each part of the problem step by step.

### Given:
- \( z_1 = -6 - 6i \)
- \( z_2 = 5 - 6i \)

### Part (a): \( 6 \cdot z_1 + 3 \cdot z_2 \)

First, calculate each term:
\[
6 \cdot z_1 = 6 \cdot (-6 - 6i) = -36 - 36i
\]
\[
3 \cdot z_2 = 3 \cdot (5 - 6i) = 15 - 18i
\]

Now, add them together:
\[
6 \cdot z_1 + 3 \cdot z_2 = (-36 - 36i) + (15 - 18i) = -21 - 54i
\]

So, the answer to part (a) is:
\[
\boxed{-21 - 54i}
\]

### Part (b): \( 4 \cdot z_1 \cdot z_2 \)

First, find the product \( z_1 \cdot z_2 \):
\[
z_1 \cdot z_2 = (-6 - 6i) \cdot (5 - 6i)
\]

Using the distributive property (FOIL):
\[
z_1 \cdot z_2 = (-6 \cdot 5) + (-6 \cdot -6i) + (-6i \cdot 5) + (-6i \cdot -6i)
\]
\[
z_1 \cdot z_2 = -30 + 36i - 30i + 36i^2
\]

Since \( i^2 = -1 \):
\[
z_1 \cdot z_2 = -30 + 6i + 36(-1) = -30 + 6i - 36 = -66 + 6i
\]

Now, multiply this by 4:
\[
4 \cdot z_1 \cdot z_2 = 4 \cdot (-66 + 6i) = -264 + 24i
\]

So, the answer to part (b) is:
\[
\boxed{-264 + 24i}
\]

### Part (c): \( \frac{z_1}{z_2} \)

To find the quotient \( \frac{z_1}{z_2} \), multiply the numerator and denominator by the conjugate of the denominator:

Let \( z_2^* = 5 + 6i \) (the conjugate of \( z_2 \)).

\[
\frac{z_1}{z_2} = \frac{-6 - 6i}{5 - 6i} \cdot \frac{5 + 6i}{5 + 6i} = \frac{(-6 - 6i)(5 + 6i)}{(5 - 6i)(5 + 6i)}
\]

First, calculate the numerator:
\[
(-6 - 6i)(5 + 6i) = -30 - 36i - 30i - 36i^2 = -30 - 66i + 36 = 6 - 66i
\]

Next, calculate the denominator:
\[
(5 - 6i)(5 + 6i) = 25 + 30i - 30i - 36i^2 = 25 + 36 = 61
\]

Thus:
\[
\frac{z_1}{z_2} = \frac{6 - 66i}{61} = \frac{6}{61} - \frac{66i}{61}
\]

So, the answer to part (c) is:
\[
\boxed{\frac{6}{61} - \frac{66i}{61}}
\]

### Summary of Answers:
a) \( -21 - 54i \)  
b) \( -264 + 24i \)  
c) \( \frac{6}{61} - \frac{66i}{61} \)

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you multiply two complex numbers?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you find the conjugate of a complex number?
4. Can you explain the process of dividing complex numbers using conjugates?
5. What are the properties of the magnitude of a complex number?

**Tip:** When multiplying complex numbers, remember to treat \(i^2\) as \(-1\) to simplify your expressions.

Explore solutions to complex number operations involving z1=-6-6i and z2=5-6i. Learn how to calculate 6⋅z1+3⋅z2, 4⋅z1⋅z2, and z1/z2 using step-by-step methods and detailed explanations. Suitable for high school students studying complex numbers.

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