Express the following complex numbers in `a+i b` form: `((3-2i)(2+3i))/((1+2i)(2-i))` (ii) `(2-sqrt(-25))/(1-sqrt(-16))`

Explanation

To express the given complex numbers in the form \( a + ib \), we will solve each part step by step.

### Part (i):
\[ \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} \]

**Step 1: Multiply the numerators and denominators.**

Numerator:
\[ (3 - 2i)(2 + 3i) = 3 \cdot 2 + 3 \cdot 3i - 2i \cdot 2 - 2i \cdot 3i \]
\[ = 6 + 9i - 4i - 6(-1) = 6 + 5i + 6 = 12 + 5i \]

Denominator:
\[ (1 + 2i)(2 - i) = 1 \cdot 2 + 1 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i) \]
\[ = 2 - i + 4i - 2(-1) = 2 - i + 4i + 2 = 4 + 3i \]

So, we have:
\[ \frac{12 + 5i}{4 + 3i} \]

**Step 2: Rationalize the denominator.**

Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(12 + 5i)(4 - 3i)}{(4 + 3i)(4 - 3i)} \]

Calculating the denominator:
\[ (4 + 3i)(4 - 3i) = 4^2 - (3i)^2 = 16 - 9(-1) = 16 + 9 = 25 \]

Calculating the numerator:
\[ (12 + 5i)(4 - 3i) = 12 \cdot 4 - 12 \cdot 3i + 5i \cdot 4 - 5i \cdot 3i \]
\[ = 48 - 36i + 20i + 15 = 63 - 16i \]

So, we have:
\[ \frac{63 - 16i}{25} \]

**Step 3: Write in the form \( a + ib \).**
\[ = \frac{63}{25} - \frac{16}{25}i \]

Thus, the final answer for part (i) is:
\[ \frac{63}{25} - \frac{16}{25}i \]

### Part (ii):
\[ \frac{2 - \sqrt{-25}}{1 - \sqrt{-16}} \]

**Step 1: Simplify the square roots.**
\[ \sqrt{-25} = 5i \quad \text{and} \quad \sqrt{-16} = 4i \]
So, we rewrite the expression as:
\[ \frac{2 - 5i}{1 - 4i} \]

**Step 2: Rationalize the denominator.**
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(2 - 5i)(1 + 4i)}{(1 - 4i)(1 + 4i)} \]

Calculating the denominator:
\[ (1 - 4i)(1 + 4i) = 1^2 - (4i)^2 = 1 - 16(-1) = 1 + 16 = 17 \]

Calculating the numerator:
\[ (2 - 5i)(1 + 4i) = 2 \cdot 1 + 2 \cdot 4i - 5i \cdot 1 - 5i \cdot 4i \]
\[ = 2 + 8i - 5i + 20 = 22 + 3i \]

So, we have:
\[ \frac{22 + 3i}{17} \]

**Step 3: Write in the form \( a + ib \).**
\[ = \frac{22}{17} + \frac{3}{17}i \]

Thus, the final answer for part (ii) is:
\[ \frac{22}{17} + \frac{3}{17}i \]

### Summary of Answers:
1. Part (i): \(\frac{63}{25} - \frac{16}{25}i\)
2. Part (ii): \(\frac{22}{17} + \frac{3}{17}i\)