Find the best answers to your questions with the help of IDNLearn.com's expert contributors. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
Sagot :
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]
### 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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
