find the product of the following complex numbers
12) $(5 - 2i)(5 + 2i)$
13) $(6 + i)(6 - i)$
14) $(7 - 8i)^2$
15) $(1 + i)^2$
16) what do you get when you square the complex number $(7 - 4i)$?
use the graph for questions 17 - 23
17) which point is $5 - 2i$?
18) which point is $-2 + i$?
19) which point is $-3 + 4i$?
20) which point is $5i$?
21) plot point k as $-5 - i$
22) plot point l as $-4$
23) plot point m as $-6i$

Question

Accepted Answer

### Problem 12: $(5 - 2i)(5 + 2i)$
# Explanation:
## Step 1: Apply the difference of squares formula $(a - b)(a + b) = a^2 - b^2$
Here, $a = 5$ and $b = 2i$. So we have $5^2 - (2i)^2$.
## Step 2: Simplify each term
Calculate $5^2 = 25$ and $(2i)^2 = 4i^2$. Since $i^2 = -1$, then $4i^2 = 4(-1) = -4$.
## Step 3: Combine the terms
Substitute back: $25 - (-4) = 25 + 4 = 29$.
# Answer:
$29$

### Problem 13: $(6 + i)(6 - i)$
# Explanation:
## Step 1: Apply the difference of squares formula $(a + b)(a - b) = a^2 - b^2$
Here, $a = 6$ and $b = i$. So we get $6^2 - i^2$.
## Step 2: Simplify each term
Calculate $6^2 = 36$ and $i^2 = -1$.
## Step 3: Combine the terms
Substitute back: $36 - (-1) = 36 + 1 = 37$.
# Answer:
$37$

### Problem 14: $(7 - 8i)^2$
# Explanation:
## Step 1: Apply the square of a binomial formula $(a - b)^2 = a^2 - 2ab + b^2$
Here, $a = 7$ and $b = 8i$. So we have $7^2 - 2(7)(8i) + (8i)^2$.
## Step 2: Simplify each term
Calculate $7^2 = 49$, $2(7)(8i) = 112i$, and $(8i)^2 = 64i^2$. Since $i^2 = -1$, $64i^2 = 64(-1) = -64$.
## Step 3: Combine the terms
Substitute back: $49 - 112i - 64 = (49 - 64) - 112i = -15 - 112i$.
# Answer:
$-15 - 112i$

### Problem 15: $(1 + i)^2$
# Explanation:
## Step 1: Apply the square of a binomial formula $(a + b)^2 = a^2 + 2ab + b^2$
Here, $a = 1$ and $b = i$. So we get $1^2 + 2(1)(i) + i^2$.
## Step 2: Simplify each term
Calculate $1^2 = 1$, $2(1)(i) = 2i$, and $i^2 = -1$.
## Step 3: Combine the terms
Substitute back: $1 + 2i - 1 = (1 - 1) + 2i = 2i$.
# Answer:
$2i$

### Problem 16: Square of $(7 - 4i)$
# Explanation:
## Step 1: Apply the square of a binomial formula $(a - b)^2 = a^2 - 2ab + b^2$
Here, $a = 7$ and $b = 4i$. So we have $7^2 - 2(7)(4i) + (4i)^2$.
## Step 2: Simplify each term
Calculate $7^2 = 49$, $2(7)(4i) = 56i$, and $(4i)^2 = 16i^2$. Since $i^2 = -1$, $16i^2 = 16(-1) = -16$.
## Step 3: Combine the terms
Substitute back: $49 - 56i - 16 = (49 - 16) - 56i = 33 - 56i$.
# Answer:
$33 - 56i$

### Problem 17: Which point is $5 - 2i$?
# Explanation:
A complex number $a + bi$ is plotted as the point $(a, b)$ on the complex plane (where the x - axis is the real axis and the y - axis is the imaginary axis). For $5 - 2i$, $a = 5$ (real part) and $b=-2$ (imaginary part). So we look for the point with x - coordinate $5$ and y - coordinate $-2$ on the given graph. (Assuming the graph has a coordinate system where we can identify such a point. If we assume the graph has points labeled, we would find the point with real part $5$ and imaginary part $-2$.)
# Answer:
(The answer would depend on the actual graph, but in general, the point corresponding to $5 - 2i$ is $(5, - 2)$ on the complex plane. If there are labeled points, we need to identify the one with x = 5 and y=-2))

### Problem 18: Which point is $-2 + i$?
# Explanation:
For a complex number $a+bi$, the coordinates on the complex plane are $(a,b)$. For $-2 + i$, $a=-2$ (real part) and $b = 1$ (imaginary part). So we look for the point with x - coordinate $-2$ and y - coordinate $1$ on the given graph.
# Answer:
(The answer would depend on the actual graph, but in general, the point corresponding to $-2 + i$ is $(-2,1)$ on the complex plane. If there are labeled points, we need to identify the one with x=-2 and y = 1))

### Problem 19: Which point is $-3 + 4i$?
# Explanation:
For the complex number $-3 + 4i$, using the complex plane representation $(a,b)$ where $a=-3$ (real) and $b = 4$ (imaginary), we look for the point with x - coordinate $-3$ and y - coordinate $4$ on the given graph.
# Answer:
(The answer would depend on the actual graph, but in general, the point corresponding to $-3 + 4i$ is $(-3,4)$ on the complex plane. If there are labeled points, we need to identify the one with x=-3 and y = 4))

### Problem 20: Which point is $5i$?
# Explanation:
The complex number $5i$ can be written as $0+5i$. So in the complex plane, $a = 0$ (real part) and $b = 5$ (imaginary part). So we look for the point with x - coordinate $0$ and y - coordinate $5$ on the given graph.
# Answer:
(The answer would depend on the actual graph, but in general, the point corresponding to $5i$ is $(0,5)$ on the complex plane. If there are labeled points, we need to identify the one with x = 0 and y = 5))

### Problem 21: Plot point $K$ as $-5 - i$
# Explanation:
For the complex number $-5 - i$, in the complex plane, $a=-5$ (real part) and $b=-1$ (imaginary part). So we plot the point with x - coordinate $-5$ and y - coordinate $-1$ on the graph.
# Answer:
(The point $K$ should be plotted at $(-5,-1)$ on the complex plane (graph))

### Problem 22: Plot point $L$ as $-4$
# Explanation:
The complex number $-4$ can be written as $-4+0i$. So in the complex plane, $a=-4$ (real part) and $b = 0$ (imaginary part). So we plot the point with x - coordinate $-4$ and y - coordinate $0$ on the graph.
# Answer:
(The point $L$ should be plotted at $(-4,0)$ on the complex plane (graph))

### Problem 23: Plot point $M$ as $-6i$
# Explanation:
The complex number $-6i$ can be written as $0-6i$. So in the complex plane, $a = 0$ (real part) and $b=-6$ (imaginary part). So we plot the point with x - coordinate $0$ and y - coordinate $-6$ on the graph.
# Answer:
(The point $M$ should be plotted at $(0,-6)$ on the complex plane (graph))

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QUESTION IMAGE

Question

Explanation:

Problem 12: \((5 - 2i)(5 + 2i)\)

Step 1: Apply the difference of squares formula \((a - b)(a + b) = a^2 - b^2\)

Step 2: Simplify each term

Step 3: Combine the terms

Step 1: Apply the difference of squares formula \((a + b)(a - b) = a^2 - b^2\)

Step 2: Simplify each term

Step 3: Combine the terms

Step 1: Apply the square of a binomial formula \((a - b)^2 = a^2 - 2ab + b^2\)

Step 2: Simplify each term

Step 3: Combine the terms

Answer:

Problem 13: \((6 + i)(6 - i)\)