Question

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`(5+2i) - (7 + i)`

`(8-i)+(3-i)`

`(-5i)(6+5i)`

`(-7i)(7i)`

`(4+4i)(3-6i)`

`(2i)(3+5i)(-5+8i)`

`(-4-7i)^2`

`(5+7i)(5-7i)`

`4/-8i`

`8+2i/7i`

Answer 1

For many of these, it is important to remember:

`i=√(-1)` ---->
`i^(2) = -1`

Answer:

`(5 + 2i) - (7+i) = 5 + 2i - 7 - i = -2 + i`

`(8-i)+(3-i) = 8-i+3-i=11-2i`

For the multiplication ones, we have to foil:

`(-5i)(6+5i) = -30i - 25i^(2) = -30i + 25`

`(-7i)(7i) = -49i^(2) = -49(-1) = 49`

`(4+4i)(3-6i)=12-24i+12i-24i^(2) = 12-12i-24(-1) = 36-12i`

`(2i)(3+5i)(-5+8i) = (6i+10i^(2) )(-5+8i) = (6i+10(-1))(-5+8i)\\=(6i-10)(-5+8i)=-30i+48i^(2) +50-80i=48(-1)+50-110i=2-110i`

`(-4-7i)^(2) = (-4-7i)(-4-7i)=16+28i+28i+49i^(2) = 16+56i-49 = 56i-33`

`(5+7i)(5-7i) = 25-35i+35i-49i^(2) = 25 -49(-1)=25+49 = 74`

`(4)/(-8i) =-(1)/(2i)=-(1)/(2i)((i)/(i))=-(i)/(2i^(2) ) =-(i)/(2(-1)) =(i)/(2)`

`(8+2i)/(7i)=(8+2i)/(7i)((i)/(i) )=(8i+2i^(2) )/(7i^(2) ) =(8i+2(-1))/(7(-1)) =(8i-2)/(-7) =(2)/(7) -(8i)/(7)`

Answer 2

See below

Explanation:

Let's go through each expression one by one:

1.
`\tt (5 + 2i) - (7 + i)`

`\tt = 5 + 2i - 7 - i`

`\tt = (5 - 7) + (2i - i)`

`\tt = -2 + i`

2.
`\tt (8 - i) + (3 - i)`

`\tt = 8 - i + 3 - i`

`\tt = (8 + 3) + (-i - i)`

`\tt = 11 - 2i`

3.
`\tt (-5i)(6 + 5i)`

`\tt = -5i \cdot 6 - 5i \cdot 5i`

`\tt = -30i - 25i^2`

Since
`\tt i^2 = -1`, replace
`\tt i^2` with
`\tt -1`:

`\tt = -30i - 25(-1)`

`\tt = -30i + 25`

4.
`\tt (-7i)(7i)`

`\tt = -7i \cdot 7i`

`\tt = -49i^2`

Since
`\tt i^2 = -1`, replace
`\tt i^2` with
`\tt -1`:

`\tt = -49(-1)`

`\tt = 49`

5.
`\tt (4 + 4i)(3 - 6i)`

`\tt = 4(3) + 4(3)(-6i) + 4i(3) + 4i(-6i)`

`\tt = 12 - 24i + 12i - 24i^2`

Since
`\tt i^2 = -1`, replace
`\tt i^2` with
`\tt -1`:

`\tt = 12 - 12i - 24(-1)`

`\tt = 36 - 12i`

6.
`\tt (2i)(3 + 5i)(-5 + 8i)`

First, multiply the first two terms:

`\tt = (2i)(3 + 5i) \cdot (-5 + 8i)`

`\tt = (6i + 10i^2)(-5 + 8i)`

`\tt = (10i^2 - 6i)(-5 + 8i)`

Replace
`\tt i^2` with
`\tt -1`:

`\tt = (-10 - 6i)(-5 + 8i)`

Now, distribute:

`\tt = 50 + 30i - 40i + 24i^2`

Replace
`\tt i^2` with
`\tt -1`:

`\tt = 50 - 10i`

7.
`\tt (-4 - 7i)^2`

`\tt = (-4 - 7i)(-4 - 7i)`

Use FOIL:

`\tt = 16 + 28i + 28i + 49i^2`

Replace
`\tt i^2` with
`\tt -1`:

`\tt = 16 + 56i - 49`

`\tt = -33 + 56i`

8.
`\tt (5 + 7i)(5 - 7i)`

Use the difference of squares formula:
`\tt a^2 - b^2 = (a + b)(a - b)`

`\tt = (5 + 7i)(5 - 7i)`

`\tt = 5^2 - (7i)^2`

`\tt = 25 - 49i^2`

Replace
`\tt i^2` with
`\tt -1`:

`\tt = 25 + 49`

`\tt = 74`

9.
`\tt (4)/(-8i)`

Multiply the numerator and denominator by
`\tt i` to eliminate the denominator:

`\tt = (4i)/(-8i^2)`

Replace
`\tt i^2` with
`\tt -1`:

`\tt = (4i)/(8)`

`\tt = (1)/(2)i`

10.
`\tt (8 + 2i)/(7i)`

Multiply the numerator and denominator by
`\tt -i` to eliminate the denominator:

`\tt = ((8 + 2i)(-i))/(7i(-i))`

`\tt = (-8i - 2i^2)/(-7)`

Replace
`\tt i^2` with
`\tt -1`:

`\tt = (-8i + 2)/(-7)`

`\tt = (2 - 8i)/(7)`