Answer :
Alright, let's analyze each of the given statements one by one to determine their truth value based on the operations with complex numbers [tex]\(a = 3 - 5i\)[/tex] and [tex]\(b = -1 + 7i\)[/tex].
### 1. Checking [tex]\(ab = 38 + 16i\)[/tex]
We need to verify if the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is equal to [tex]\(38 + 16i\)[/tex].
Given [tex]\(a = 3 - 5i\)[/tex] and [tex]\(b = -1 + 7i\)[/tex], the product [tex]\(ab = (3 - 5i)(-1 + 7i)\)[/tex].
Upon performing multiplication:
[tex]\(ab = 3(-1) + 3(7i) - 5i(-1) - 5i(7i)\)[/tex]
[tex]\(ab = -3 + 21i + 5i - 35i^2\)[/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\(ab = -3 + 21i + 5i + 35\)[/tex]
[tex]\(ab = -3 + 21i + 5i + 35\)[/tex]
[tex]\(ab = 32 + 26i\)[/tex]
Thus, [tex]\(ab = 32 + 26i\)[/tex].
So, [tex]\(ab \neq 38 + 16i\)[/tex]. This statement is false.
### 2. Checking [tex]\(ab = 32 + 26i\)[/tex]
As calculated above, the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is indeed [tex]\(32 + 26i\)[/tex].
This statement is true.
### 3. Checking [tex]\(\frac{a}{b} = \frac{19 - 81}{25}\)[/tex]
We need to check the division of [tex]\(a\)[/tex] by [tex]\(b\)[/tex].
Given [tex]\(a = 3 - 5i\)[/tex] and [tex]\(b = -1 + 7i\)[/tex], let's calculate [tex]\(\frac{a}{b}\)[/tex]:
[tex]\[ \frac{a}{b} = \frac{3 - 5i}{-1 + 7i} \][/tex]
To simplify, we multiply the numerator and denominator by the complex conjugate of the denominator [tex]\(-1 - 7i\)[/tex]:
[tex]\[ \frac{a}{b} = \frac{(3 - 5i)(-1 - 7i)}{(-1 + 7i)(-1 - 7i)} \][/tex]
The denominator [tex]\((-1 + 7i)(-1 - 7i) = 1 + 49 = 50\)[/tex].
The numerator:
[tex]\[ (3 - 5i)(-1 - 7i) = 3(-1) + 3(-7i) - 5i(-1) - 5i(-7i) \][/tex]
[tex]\[ = -3 - 21i + 5i + 35i^2 \][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[ = -3 - 21i + 5i - 35 \][/tex]
[tex]\[ = -38 - 16i \][/tex]
[tex]\[ \frac{a}{b} = \frac{-38 - 16i}{50} = -\frac{38}{50} - \frac{16i}{50} = -\frac{19}{25} - \frac{8i}{25} \][/tex]
So, [tex]\(\frac{a}{b} \neq \frac{19 - 8i}{25}\)[/tex]. This statement is incorrectly written. It should be [tex]\(\frac{a}{b} = -\frac{19 + 8i}{25}\)[/tex]. The provided statement is false.
### 4. Checking [tex]\(ab = 2 + 2i\)[/tex]
As calculated above, [tex]\(ab ≠ 2 + 2i\)[/tex].
This statement is false.
### 5. Checking [tex]\(a = \frac{21 + 51}{7}\)[/tex]
This statement does not make sense because [tex]\(a\)[/tex] is a complex number, and the given format suggests a real number expression.
This statement is false.
### 6. Checking [tex]\(\frac{a}{b} = -\frac{19 + 8i}{25}\)[/tex]
As we calculated above:
[tex]\[ \frac{a}{b} = -\frac{19}{25} - \frac{8i}{25} \][/tex]
This matches the provided statement.
This statement is true.
### Summary of true statements:
- [tex]\(ab = 32 + 26i\)[/tex]
- [tex]\(\frac{a}{b} = -\frac{19 + 8i}{25}\)[/tex]
These are the true statements based on our verification:
[tex]\[ ab = 32 + 26i \][/tex]
[tex]\[ \frac{a}{b} = -\frac{19 + 8i}{25} \][/tex]
### 1. Checking [tex]\(ab = 38 + 16i\)[/tex]
We need to verify if the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is equal to [tex]\(38 + 16i\)[/tex].
Given [tex]\(a = 3 - 5i\)[/tex] and [tex]\(b = -1 + 7i\)[/tex], the product [tex]\(ab = (3 - 5i)(-1 + 7i)\)[/tex].
Upon performing multiplication:
[tex]\(ab = 3(-1) + 3(7i) - 5i(-1) - 5i(7i)\)[/tex]
[tex]\(ab = -3 + 21i + 5i - 35i^2\)[/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\(ab = -3 + 21i + 5i + 35\)[/tex]
[tex]\(ab = -3 + 21i + 5i + 35\)[/tex]
[tex]\(ab = 32 + 26i\)[/tex]
Thus, [tex]\(ab = 32 + 26i\)[/tex].
So, [tex]\(ab \neq 38 + 16i\)[/tex]. This statement is false.
### 2. Checking [tex]\(ab = 32 + 26i\)[/tex]
As calculated above, the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is indeed [tex]\(32 + 26i\)[/tex].
This statement is true.
### 3. Checking [tex]\(\frac{a}{b} = \frac{19 - 81}{25}\)[/tex]
We need to check the division of [tex]\(a\)[/tex] by [tex]\(b\)[/tex].
Given [tex]\(a = 3 - 5i\)[/tex] and [tex]\(b = -1 + 7i\)[/tex], let's calculate [tex]\(\frac{a}{b}\)[/tex]:
[tex]\[ \frac{a}{b} = \frac{3 - 5i}{-1 + 7i} \][/tex]
To simplify, we multiply the numerator and denominator by the complex conjugate of the denominator [tex]\(-1 - 7i\)[/tex]:
[tex]\[ \frac{a}{b} = \frac{(3 - 5i)(-1 - 7i)}{(-1 + 7i)(-1 - 7i)} \][/tex]
The denominator [tex]\((-1 + 7i)(-1 - 7i) = 1 + 49 = 50\)[/tex].
The numerator:
[tex]\[ (3 - 5i)(-1 - 7i) = 3(-1) + 3(-7i) - 5i(-1) - 5i(-7i) \][/tex]
[tex]\[ = -3 - 21i + 5i + 35i^2 \][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[ = -3 - 21i + 5i - 35 \][/tex]
[tex]\[ = -38 - 16i \][/tex]
[tex]\[ \frac{a}{b} = \frac{-38 - 16i}{50} = -\frac{38}{50} - \frac{16i}{50} = -\frac{19}{25} - \frac{8i}{25} \][/tex]
So, [tex]\(\frac{a}{b} \neq \frac{19 - 8i}{25}\)[/tex]. This statement is incorrectly written. It should be [tex]\(\frac{a}{b} = -\frac{19 + 8i}{25}\)[/tex]. The provided statement is false.
### 4. Checking [tex]\(ab = 2 + 2i\)[/tex]
As calculated above, [tex]\(ab ≠ 2 + 2i\)[/tex].
This statement is false.
### 5. Checking [tex]\(a = \frac{21 + 51}{7}\)[/tex]
This statement does not make sense because [tex]\(a\)[/tex] is a complex number, and the given format suggests a real number expression.
This statement is false.
### 6. Checking [tex]\(\frac{a}{b} = -\frac{19 + 8i}{25}\)[/tex]
As we calculated above:
[tex]\[ \frac{a}{b} = -\frac{19}{25} - \frac{8i}{25} \][/tex]
This matches the provided statement.
This statement is true.
### Summary of true statements:
- [tex]\(ab = 32 + 26i\)[/tex]
- [tex]\(\frac{a}{b} = -\frac{19 + 8i}{25}\)[/tex]
These are the true statements based on our verification:
[tex]\[ ab = 32 + 26i \][/tex]
[tex]\[ \frac{a}{b} = -\frac{19 + 8i}{25} \][/tex]
