Answer :
Alright! Let's go through each pair of complex numbers step-by-step to find their sums, differences, products, and quotients.
### Pair (a): [tex]\( 2 + 5i \)[/tex] and [tex]\( 7 - i \)[/tex]
#### Sum:
[tex]\[ (2 + 5i) + (7 - i) = 2 + 7 + (5i - i) = 9 + 4i \][/tex]
#### Difference:
[tex]\[ (2 + 5i) - (7 - i) = 2 - 7 + (5i + i) = -5 + 6i \][/tex]
#### Product:
[tex]\[ (2 + 5i) \cdot (7 - i) = 2 \cdot 7 + 2 \cdot (-i) + 5i \cdot 7 + 5i \cdot (-i) = 14 - 2i + 35i - 5i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 14 - 2i + 35i - 5(-1) = 14 - 2i + 35i + 5 = 19 + 33i \][/tex]
#### Quotient:
[tex]\[ \frac{2 + 5i}{7 - i} = (2 + 5i) \cdot \frac{7 + i}{7 + i} = \frac{(2 + 5i)(7 + i)}{(7 - i)(7 + i)} \][/tex]
[tex]\[ (7 - i)(7 + i) = 7^2 - i^2 = 49 + 1 = 50 \quad \text{(since } i^2 = -1) \][/tex]
[tex]\[ (2 + 5i)(7 + i) = 14 + 2i + 35i + 5i^2 = 14 + 37i - 5 = 9 + 37i \quad \text{(since } 5i^2 = -5) \][/tex]
Thus:
[tex]\[ \frac{9 + 37i}{50} = 0.18 + 0.74i \][/tex]
### Pair (b): [tex]\( 4 - i \)[/tex] and [tex]\( 3 - 3i \)[/tex]
#### Sum:
[tex]\[ (4 - i) + (3 - 3i) = 4 + 3 + (-i - 3i) = 7 - 4i \][/tex]
#### Difference:
[tex]\[ (4 - i) - (3 - 3i) = 4 - 3 + (-i + 3i) = 1 + 2i \][/tex]
#### Product:
[tex]\[ (4 - i) \cdot (3 - 3i) = 4 \cdot 3 + 4 \cdot (-3i) - i \cdot 3 - i \cdot (-3i) = 12 - 12i - 3i + 3i^2 = 12 - 15i + 3(-1) \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 12 - 15i - 3 = 9 - 15i \][/tex]
#### Quotient:
[tex]\[ \frac{4 - i}{3 - 3i} = (4 - i) \cdot \frac{3 + 3i}{3 + 3i} = \frac{(4 - i)(3 + 3i)}{(3 - 3i)(3 + 3i)} \][/tex]
[tex]\[ (3 - 3i)(3 + 3i) = 3^2 - (3i)^2 = 9 + 9 = 18 \quad \text{(since } (3i)^2 = 9i^2 = 9(-1) = -9) \][/tex]
[tex]\[ (4 - i)(3 + 3i) = 12 + 12i - 3i - 3i^2 = 12 + 9i + 3(-1) \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 12 + 9i - 3 = 9 + 9i \][/tex]
Thus:
[tex]\[ \frac{9 + 9i}{18} = 0.5 + 0.5i = 0.8333 + 0.5i \][/tex]
### Pair (c): [tex]\( 2 + 7i \)[/tex] and [tex]\( 4 - 9i \)[/tex]
#### Sum:
[tex]\[ (2 + 7i) + (4 - 9i) = 2 + 4 + (7i - 9i) = 6 - 2i \][/tex]
#### Difference:
[tex]\[ (2 + 7i) - (4 - 9i) = 2 - 4 + (7i + 9i) = -2 + 16i \][/tex]
#### Product:
[tex]\[ (2 + 7i) \cdot (4 - 9i) = 2 \cdot 4 + 2 \cdot (-9i) + 7i \cdot 4 + 7i \cdot (-9i) = 8 - 18i + 28i - 63i^2 = 8 + 10i - 63(-1) \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 8 + 10i + 63 = 71 + 10i \][/tex]
#### Quotient:
[tex]\[ \frac{2 + 7i}{4 - 9i} = (2 + 7i) \cdot \frac{4 + 9i}{4 + 9i} = \frac{(2 + 7i)(4 + 9i)}{(4 - 9i)(4 + 9i)} \][/tex]
[tex]\[ (4 - 9i)(4 + 9i) = 4^2 - (9i)^2 = 16 + 81 = 97 \quad \text{(since } 81i^2 = 81(-1) = -81) \][/tex]
[tex]\[ (2 + 7i)(4 + 9i) = 8 + 18i + 28i + 63i^2 = 8 + 46i - 63 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 8 + 46i - 63 = -55 + 46i \][/tex]
Thus:
[tex]\[ \frac{-55 + 46i}{97} = -0.5670 + 0.4742i \][/tex]
### Pair (d): [tex]\( a + bi \)[/tex] and [tex]\( c - di \)[/tex]
To find the sum, difference, product, and quotient with these general expressions of complex numbers:
#### Sum:
[tex]\[ (a + bi) + (c - di) = (a + c) + (b - d)i \][/tex]
#### Difference:
[tex]\[ (a + bi) - (c - di) = (a - c) + (b + d)i \][/tex]
#### Product:
[tex]\[ (a + bi) \cdot (c - di) = a \cdot c + a \cdot (-di) + bi \cdot c + bi \cdot (-di) = ac - adi + bci - bdi^2 = ac - adi + bci + bd \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ = ac - adi + bci + bd(-1) = ac - adi + bci - bd \][/tex]
Thus,
[tex]\[ (ac - bd) + (bc - ad)i \][/tex]
#### Quotient:
[tex]\[ \frac{a + bi}{c - di} = \frac{(a + bi) \cdot (c + di)}{(c - di) \cdot (c + di)} = \frac{ac + adi + bci + bdi^2}{c^2 + d^2} \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ = \frac{ac + adi + bci + bd(-1)}{c^2 + d^2} = \frac{ac + adi + bci - bd}{c^2 + d^2} \][/tex]
Thus,
[tex]\[ \frac{ac - bd + (bc + ad)i}{c^2 + d^2} = \frac{ac - bd}{c^2 + d^2} + \frac{bc + ad}{c^2 + d^2}i \][/tex]
### Pair (a): [tex]\( 2 + 5i \)[/tex] and [tex]\( 7 - i \)[/tex]
#### Sum:
[tex]\[ (2 + 5i) + (7 - i) = 2 + 7 + (5i - i) = 9 + 4i \][/tex]
#### Difference:
[tex]\[ (2 + 5i) - (7 - i) = 2 - 7 + (5i + i) = -5 + 6i \][/tex]
#### Product:
[tex]\[ (2 + 5i) \cdot (7 - i) = 2 \cdot 7 + 2 \cdot (-i) + 5i \cdot 7 + 5i \cdot (-i) = 14 - 2i + 35i - 5i^2 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 14 - 2i + 35i - 5(-1) = 14 - 2i + 35i + 5 = 19 + 33i \][/tex]
#### Quotient:
[tex]\[ \frac{2 + 5i}{7 - i} = (2 + 5i) \cdot \frac{7 + i}{7 + i} = \frac{(2 + 5i)(7 + i)}{(7 - i)(7 + i)} \][/tex]
[tex]\[ (7 - i)(7 + i) = 7^2 - i^2 = 49 + 1 = 50 \quad \text{(since } i^2 = -1) \][/tex]
[tex]\[ (2 + 5i)(7 + i) = 14 + 2i + 35i + 5i^2 = 14 + 37i - 5 = 9 + 37i \quad \text{(since } 5i^2 = -5) \][/tex]
Thus:
[tex]\[ \frac{9 + 37i}{50} = 0.18 + 0.74i \][/tex]
### Pair (b): [tex]\( 4 - i \)[/tex] and [tex]\( 3 - 3i \)[/tex]
#### Sum:
[tex]\[ (4 - i) + (3 - 3i) = 4 + 3 + (-i - 3i) = 7 - 4i \][/tex]
#### Difference:
[tex]\[ (4 - i) - (3 - 3i) = 4 - 3 + (-i + 3i) = 1 + 2i \][/tex]
#### Product:
[tex]\[ (4 - i) \cdot (3 - 3i) = 4 \cdot 3 + 4 \cdot (-3i) - i \cdot 3 - i \cdot (-3i) = 12 - 12i - 3i + 3i^2 = 12 - 15i + 3(-1) \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 12 - 15i - 3 = 9 - 15i \][/tex]
#### Quotient:
[tex]\[ \frac{4 - i}{3 - 3i} = (4 - i) \cdot \frac{3 + 3i}{3 + 3i} = \frac{(4 - i)(3 + 3i)}{(3 - 3i)(3 + 3i)} \][/tex]
[tex]\[ (3 - 3i)(3 + 3i) = 3^2 - (3i)^2 = 9 + 9 = 18 \quad \text{(since } (3i)^2 = 9i^2 = 9(-1) = -9) \][/tex]
[tex]\[ (4 - i)(3 + 3i) = 12 + 12i - 3i - 3i^2 = 12 + 9i + 3(-1) \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 12 + 9i - 3 = 9 + 9i \][/tex]
Thus:
[tex]\[ \frac{9 + 9i}{18} = 0.5 + 0.5i = 0.8333 + 0.5i \][/tex]
### Pair (c): [tex]\( 2 + 7i \)[/tex] and [tex]\( 4 - 9i \)[/tex]
#### Sum:
[tex]\[ (2 + 7i) + (4 - 9i) = 2 + 4 + (7i - 9i) = 6 - 2i \][/tex]
#### Difference:
[tex]\[ (2 + 7i) - (4 - 9i) = 2 - 4 + (7i + 9i) = -2 + 16i \][/tex]
#### Product:
[tex]\[ (2 + 7i) \cdot (4 - 9i) = 2 \cdot 4 + 2 \cdot (-9i) + 7i \cdot 4 + 7i \cdot (-9i) = 8 - 18i + 28i - 63i^2 = 8 + 10i - 63(-1) \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 8 + 10i + 63 = 71 + 10i \][/tex]
#### Quotient:
[tex]\[ \frac{2 + 7i}{4 - 9i} = (2 + 7i) \cdot \frac{4 + 9i}{4 + 9i} = \frac{(2 + 7i)(4 + 9i)}{(4 - 9i)(4 + 9i)} \][/tex]
[tex]\[ (4 - 9i)(4 + 9i) = 4^2 - (9i)^2 = 16 + 81 = 97 \quad \text{(since } 81i^2 = 81(-1) = -81) \][/tex]
[tex]\[ (2 + 7i)(4 + 9i) = 8 + 18i + 28i + 63i^2 = 8 + 46i - 63 \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ 8 + 46i - 63 = -55 + 46i \][/tex]
Thus:
[tex]\[ \frac{-55 + 46i}{97} = -0.5670 + 0.4742i \][/tex]
### Pair (d): [tex]\( a + bi \)[/tex] and [tex]\( c - di \)[/tex]
To find the sum, difference, product, and quotient with these general expressions of complex numbers:
#### Sum:
[tex]\[ (a + bi) + (c - di) = (a + c) + (b - d)i \][/tex]
#### Difference:
[tex]\[ (a + bi) - (c - di) = (a - c) + (b + d)i \][/tex]
#### Product:
[tex]\[ (a + bi) \cdot (c - di) = a \cdot c + a \cdot (-di) + bi \cdot c + bi \cdot (-di) = ac - adi + bci - bdi^2 = ac - adi + bci + bd \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ = ac - adi + bci + bd(-1) = ac - adi + bci - bd \][/tex]
Thus,
[tex]\[ (ac - bd) + (bc - ad)i \][/tex]
#### Quotient:
[tex]\[ \frac{a + bi}{c - di} = \frac{(a + bi) \cdot (c + di)}{(c - di) \cdot (c + di)} = \frac{ac + adi + bci + bdi^2}{c^2 + d^2} \][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[ = \frac{ac + adi + bci + bd(-1)}{c^2 + d^2} = \frac{ac + adi + bci - bd}{c^2 + d^2} \][/tex]
Thus,
[tex]\[ \frac{ac - bd + (bc + ad)i}{c^2 + d^2} = \frac{ac - bd}{c^2 + d^2} + \frac{bc + ad}{c^2 + d^2}i \][/tex]
