The problem is to solve the complex number arithmetic problems shown in the image. Specifically, we will solve questions 24, 28, 29, 30 and 31. 24. $3i(2+2i)$ 25. $\frac{5+2i}{4i}$ 26. $\frac{3i}{-2+i}$ 27. $\frac{3-2i}{4-3i}$ 28. $\frac{7}{5-2i}$

1. Problem Description

The problem is to solve the complex number arithmetic problems shown in the image. Specifically, we will solve questions 24, 28, 29, 30 and

3

1.

2

4. $3i(2+2i)$

2

5. $\frac{5+2i}{4i}$

2

6. $\frac{3i}{-2+i}$

2

7. $\frac{3-2i}{4-3i}$

2

8. $\frac{7}{5-2i}$

2. Solution Steps

Problem 24: $3i(2+2i)$

We distribute the $3i$ across the terms in the parentheses:

$3i(2) + 3i(2i) = 6i + 6i^2$

Since $i^2 = -1$, we have:

$6i + 6(-1) = -6 + 6i$

Problem 28: $\frac{5+2i}{4i}$

To get rid of $i$ in the denominator, multiply the numerator and denominator by the conjugate of $4i$, which is $-4i$:

$\frac{5+2i}{4i} \cdot \frac{-4i}{-4i} = \frac{-20i - 8i^2}{-16i^2} = \frac{-20i - 8(-1)}{-16(-1)} = \frac{8 - 20i}{16}$

Divide both terms in numerator by 8, and denominator by 8 to get:

$\frac{1 - \frac{5}{2}i}{2} = \frac{1}{2} - \frac{5}{2}i$

Problem 29: $\frac{3i}{-2+i}$

Multiply the numerator and denominator by the conjugate of $-2+i$, which is $-2-i$:

$\frac{3i}{-2+i} \cdot \frac{-2-i}{-2-i} = \frac{-6i - 3i^2}{4 - i^2} = \frac{-6i - 3(-1)}{4 - (-1)} = \frac{3-6i}{5} = \frac{3}{5} - \frac{6}{5}i$

Problem 30: $\frac{3-2i}{4-3i}$

Multiply the numerator and denominator by the conjugate of $4-3i$, which is $4+3i$:

$\frac{3-2i}{4-3i} \cdot \frac{4+3i}{4+3i} = \frac{(3-2i)(4+3i)}{(4-3i)(4+3i)} = \frac{12 + 9i - 8i - 6i^2}{16 - 9i^2} = \frac{12+i-6(-1)}{16-9(-1)} = \frac{12+i+6}{16+9} = \frac{18+i}{25} = \frac{18}{25} + \frac{1}{25}i$

Problem 31: $\frac{7}{5-2i}$

Multiply the numerator and denominator by the conjugate of $5-2i$, which is $5+2i$:

$\frac{7}{5-2i} \cdot \frac{5+2i}{5+2i} = \frac{7(5+2i)}{(5-2i)(5+2i)} = \frac{35+14i}{25 - 4i^2} = \frac{35+14i}{25-4(-1)} = \frac{35+14i}{25+4} = \frac{35+14i}{29} = \frac{35}{29} + \frac{14}{29}i$

3. Final Answer

Problem 24:

$-6 + 6i$

Problem 28:

$\frac{1}{2} - \frac{5}{2}i$

Problem 29:

$\frac{3}{5} - \frac{6}{5}i$

Problem 30:

$\frac{18}{25} + \frac{1}{25}i$

Problem 31:

$\frac{35}{29} + \frac{14}{29}i$