Sorry the first post formatted weirdly.

1. 3i - 10
---------
8i - 5

Answer 74 65i
---- + -----
89 89

2. 11i^2 + 1
------------
-12i - 8

Answer: 5 (2 - 3 i)
-------------
26

3. square root -25 * 8 square root -49

Answer: -280

4. square root -64 + 12 square root -36

Answer: 80 i

5. square root -13 - 6 square root -13

Answer: -5 i square root 13

ok it wont let me put spaces, sorry!

1. 3i - 10 over 8i - 5

Answer: 74 over 89 + 65 i over 89

2. 11i^2 + 1 over -12i - 8

Answer: 5 (2 - 3 i) over 26

all correct good work.

You can also check your answers by typing in the expression at wolframalpha.com

To solve these complex arithmetic expressions involving imaginary numbers, follow these steps:

1. Simplify the expression by combining like terms or applying basic operations such as addition, subtraction, multiplication, and division.
2. Use the properties of square roots and imaginary numbers to simplify further.
3. If necessary, rationalize the denominator to remove any imaginary parts.

Let's go through the solutions step by step:

1. To simplify the expression (3i - 10) / (8i - 5):

Start by multiplying the numerator and denominator by the conjugate of the denominator, which is (8i + 5), to eliminate the imaginary terms in the denominator:

((3i - 10) * (8i + 5)) / ((8i - 5) * (8i + 5)) = (24i^2 + 15i - 80i - 50) / (64i^2 + 40i - 40i - 25) = (24(-1) + 15i - 80i - 50) / (64(-1) - 25)

Simplify the expression further:

(-24 - 65i) / (-64 - 25) = (-24 - 65i) / (-89)

Divide each term by -89:

-24 / -89 - (65i / -89) = 24/89 + (65i/89) = (24 + 65i) / 89

Therefore, the answer is (24 + 65i) / 89.

2. To simplify the expression (11i^2 + 1) / (-12i - 8):

Substitute i^2 with -1 in the numerator:

(11(-1) + 1) / (-12i - 8) = (-11 + 1) / (-12i - 8) = -10 / (-12i - 8) = 10 / (12i + 8)

Divide numerator and denominator by 2 for simplification:

10/2 / (12i/2 + 8/2) = 5 / (6i + 4)

Now, the denominator can be further simplified:

5 / (2(3i + 2)) = 5 / (3i + 2)

Multiply both the numerator and denominator by the conjugate of the denominator, which is (3i - 2), to rationalize the denominator:

(5 * (3i - 2)) / ((3i + 2) * (3i - 2)) = (15i - 10) / (9i^2 - 4) = (15i - 10) / (9(-1) - 4) = (15i - 10) / (-9 - 4) = (15i - 10) / (-13)

Divide each term by -13:

-15i/13 + (10/13) = (10 - 15i) / 13

Therefore, the answer is (10 - 15i) / 13.

3. To simplify the expression √(-25) * 8 * √(-49):

Simplify the square roots:

√(-25) * 8 * √(-49) = √(25 * (-1)) * 8 * √(49 * (-1)) = √(-1) * 8 * √(-1) * 7 * √(-1)

√(-1) can be represented as i:

i * 8 * i * 7 * i = -56i^3

Simplify i^3 as i * i * i = -i:

-56i^3 = -56(-i) = 56i

Therefore, the answer is 56i.

4. To simplify the expression √(-64) + 12 * √(-36):

Simplify the square roots:

√(-64) + 12 * √(-36) = √(64 * (-1)) + 12 * √(36 * (-1)) = √(-1) * √(64) + 12 * √(-1) * √(36)

√(-1) can be represented as i:

i * √(64) + 12 * i * √(36) = i * 8 + 12i * 6 = 8i + 72i = 80i

Therefore, the answer is 80i.

5. To simplify the expression √(-13) - 6 * √(-13):

Simplify the square roots:

√(-13) - 6 * √(-13) = √(13 * (-1)) - 6 * √(13 * (-1)) = √(-1) * √(13) - 6 * √(-1) * √(13)

√(-1) can be represented as i:

i * √(13) - 6 * i * √(13) = i * √(13) - 6i * √(13) = (1 - 6) * i * √(13) = -5i * √(13) = -5i√(13)

Therefore, the answer is -5i√(13).