Let's carefully analyze and simplify each given quotient to express them in the form [tex]\( a + bi \)[/tex] and match them with their provided answers:
### Quotient 1: [tex]\(\frac{1}{3 - 4i}\)[/tex]
#### Solution:
[tex]\[
\frac{1}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i} = \frac{3 + 4i}{(3 - 4i)(3 + 4i)}
\][/tex]
[tex]\[
(3 - 4i)(3 + 4i) = 9 + 12i - 12i - 16i^2 = 9 + 16 = 25
\][/tex]
So,
[tex]\[
\frac{3 + 4i}{25} = \frac{3}{25} + \frac{4i}{25}
\][/tex]
Thus,
[tex]\[
\frac{1}{3-4i} = 0.12 + 0.16i
\][/tex]
### Quotient 2: [tex]\(\frac{3 - 4i}{1}\)[/tex]
#### Solution:
This is straightforward as division by 1 leaves the complex number unchanged,
[tex]\[
\frac{3 - 4i}{1} = 3 - 4i
\][/tex]
### Quotient 3: [tex]\(\frac{i}{i}\)[/tex]
#### Solution:
Since [tex]\(i\)[/tex] divided by [tex]\(i\)[/tex] is 1,
[tex]\[
\frac{i}{i} = 1
\][/tex]
### Quotient 4: [tex]\(\frac{3 + 4i}{3 - 4i}\)[/tex]
#### Solution:
[tex]\[
\frac{3 + 4i}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i} = \frac{(3 + 4i)^2}{(3 - 4i)(3 + 4i)}
\][/tex]
[tex]\[
(3 + 4i)^2 = 9 + 24i - 16 = -7 + 24i
\][/tex]
[tex]\[
(3 - 4i)(3 + 4i) = 9 + 16 = 25
\][/tex]
So,
[tex]\[
\frac{-7 + 24i}{25} = -0.28 + 0.96i
\][/tex]
### Quotient 5: [tex]\(\frac{3 - 4i}{3 + 4i}\)[/tex]
#### Solution:
[tex]\[
\frac{3 - 4i}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} = \frac{(3 - 4i)^2}{(3 + 4i)(3 - 4i)}
\][/tex]
[tex]\[
(3 - 4i)^2 = 9 - 24i - 16 = -7 - 24i
\][/tex]
[tex]\[
(3 + 4i)(3 - 4i) = 9 + 16 = 25
\][/tex]
So,
[tex]\[
\frac{-7 - 24i}{25} = -0.28 - 0.96i
\][/tex]
### Quotient 6: [tex]\(\frac{1}{3 + 4i}\)[/tex]
#### Solution:
[tex]\[
\frac{1}{3 + 4i} \cdot \frac{3 - 4i}{3 - 4i} = \frac{3 - 4i}{(3 + 4i)(3 - 4i)}
\][/tex]
[tex]\[
(3 + 4i)(3 - 4i) = 9 + 16 = 25
\][/tex]
So,
[tex]\[
\frac{3 - 4i}{25} = \frac{3}{25} - \frac{4i}{25} = 0.12 - 0.16i
\][/tex]
### Matching the results:
1. [tex]\(\frac{1}{3-4i}\)[/tex] matches [tex]\( 0.12 + 0.16i \)[/tex]
2. [tex]\(\frac{3-4i}{1}\)[/tex] matches [tex]\( 3 - 4i \)[/tex]
3. [tex]\(\frac{i}{i}\)[/tex] matches [tex]\( 1 + 0i \)[/tex]
4. [tex]\(\frac{3+4i}{3-4i}\)[/tex] matches [tex]\( -0.28 + 0.96i \)[/tex]
5. [tex]\(\frac{3-4i}{3+4i}\)[/tex] matches [tex]\( -0.28 - 0.96i \)[/tex]
6. [tex]\(\frac{1}{3+4i}\)[/tex] matches [tex]\( 0.12 - 0.16i \)[/tex]
Thus, the correct matches are:
- [tex]\(\frac{1}{3-4i}\)[/tex] matches [tex]\( \frac{3}{25} + \frac{4}{25}i \)[/tex]
- [tex]\(\frac{3-4i}{1}\)[/tex] matches [tex]\( 3-4i \)[/tex]
- [tex]\(\frac{i}{i}\)[/tex] matches [tex]\( 1+0i \)[/tex]
- [tex]\(\frac{3+4i}{3-4i}\)[/tex] matches [tex]\( \frac{-7}{25} + \frac{24}{25}i \)[/tex]
- [tex]\(\frac{3-4i}{3+4i}\)[/tex] matches [tex]\( \frac{-7}{25} - \frac{24}{25}i \)[/tex]
- [tex]\(\frac{1}{3+4i}\)[/tex] matches [tex]\( \frac{3}{25} - \frac{4}{25}i \)[/tex]
