10) $-5 - 12i(6 + 2i)$
11) $-8i^2 - 5(2 + 3i)$
plot each complex number
12) $a = -3 - 2i$
13) $b = 7 + i$
14) $c = 6 - 8i$
15) $d = -7i$
16) $e = -8$
what complex number does each point represent?
17) $f$
18) $g$
19) $h$
20) $i$

Question

Accepted Answer

### Problem 10: Simplify $-5 - 12i(6 + 2i)$
# Explanation:
## Step 1: Distribute the $ -12i $
Using the distributive property $ a(b + c) = ab + ac $, we have:
$ -5 - 12i \times 6 - 12i \times 2i $
$ = -5 - 72i - 24i^2 $

## Step 2: Substitute $ i^2 = -1 $
Since $ i^2 = -1 $, we substitute that in:
$ -5 - 72i - 24(-1) $
$ = -5 - 72i + 24 $

## Step 3: Combine like terms
Combine the real parts ($ -5 $ and $ 24 $):
$ (-5 + 24) - 72i $
$ = 19 - 72i $

# Answer: $ 19 - 72i $

### Problem 11: Simplify $ -8i^2 - 5(2 + 3i) $
# Explanation:
## Step 1: Substitute $ i^2 = -1 $ and distribute $ -5 $
First, substitute $ i^2 = -1 $ into $ -8i^2 $, and distribute $ -5 $ in $ -5(2 + 3i) $:
$ -8(-1) - 5 \times 2 - 5 \times 3i $
$ = 8 - 10 - 15i $

## Step 2: Combine like terms
Combine the real parts ($ 8 $ and $ -10 $):
$ (8 - 10) - 15i $
$ = -2 - 15i $

# Answer: $ -2 - 15i $

### Problems 12 - 16: Plotting Complex Numbers
A complex number $ z = a + bi $ is plotted in the complex plane where the x - axis (real axis) represents the real part $ a $ and the y - axis (imaginary axis) represents the imaginary part $ b $.

#### 12) $ A=-3 - 2i $
- Real part $ a=-3 $, Imaginary part $ b = - 2 $. So we plot the point $(-3,-2)$ in the complex plane (3 units to the left of the origin on the real axis and 2 units down on the imaginary axis).

#### 13) $ B = 7 + i $
- Real part $ a = 7 $, Imaginary part $ b=1 $. Plot the point $(7,1)$ (7 units to the right of the origin on the real axis and 1 unit up on the imaginary axis).

#### 14) $ C=6 - 8i $
- Real part $ a = 6 $, Imaginary part $ b=-8 $. Plot the point $(6, - 8)$ (6 units to the right of the origin on the real axis and 8 units down on the imaginary axis).

#### 15) $ D=-7i $
- This can be written as $ D = 0-7i $, so real part $ a = 0 $, Imaginary part $ b=-7 $. Plot the point $(0,-7)$ (on the imaginary axis, 7 units down from the origin).

#### 16) $ E=-8 $
- This can be written as $ E=-8 + 0i $, so real part $ a=-8 $, Imaginary part $ b = 0 $. Plot the point $(-8,0)$ (on the real axis, 8 units to the left of the origin).

### Problems 17 - 20: Determining Complex Numbers from Points
To determine the complex number $ z=a + bi $ from a point $(x,y)$ in the complex plane, $ a=x $ (real part) and $ b = y $ (imaginary part).

#### 17) For point $ F $ (assuming from the diagram, let's say $ F=(x_F,y_F) $)
- If we assume from the diagram that $ F=(4,-1) $ (you need to check the grid), then the complex number $ F = 4 - i $.

#### 18) For point $ G $ (assuming $ G=(0,6) $ from the diagram)
- Then $ G=0 + 6i=6i $.

#### 19) For point $ H $ (assuming $ H=(-5,-1) $ from the diagram)
- Then $ H=-5 - i $.

#### 20) For point $ I $ (assuming $ I=(9,0) $ from the diagram)
- Then $ I=9+0i = 9$.

(Note: For the plotting and point - to - complex - number conversion, the exact coordinates depend on the scale of the provided grid. The above is a general method based on the complex plane representation of complex numbers.)

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QUESTION IMAGE

Question

Explanation:

Problem 10: Simplify \(-5 - 12i(6 + 2i)\)

Step 1: Distribute the \( -12i \)

Step 2: Substitute \( i^2 = -1 \)

Step 3: Combine like terms

Step 1: Substitute \( i^2 = -1 \) and distribute \( -5 \)

Step 2: Combine like terms

Answer:

Problem 11: Simplify \( -8i^2 - 5(2 + 3i) \)