The problem asks us to simplify several expressions involving complex numbers. We need to perform addition, subtraction, multiplication, and squaring operations on these expressions, remembering that $i = \sqrt{-1}$ and $i^2 = -1$. We also need to evaluate square roots of negative numbers.

1. Problem Description

The problem asks us to simplify several expressions involving complex numbers. We need to perform addition, subtraction, multiplication, and squaring operations on these expressions, remembering that $i = \sqrt{-1}$ and $i^2 = -1$. We also need to evaluate square roots of negative numbers.

2. Solution Steps

1

0. $(-2+3i) + (5-2i)$

Combine the real and imaginary parts:

$(-2+5) + (3i - 2i) = 3 + i$

1

1. $(-6+7i) + (6-7i)$

Combine the real and imaginary parts:

$(-6+6) + (7i - 7i) = 0 + 0i = 0$

1

2. $(4-2i) - (-1+3i)$

Distribute the negative sign and combine terms:

$4-2i + 1 - 3i = (4+1) + (-2i - 3i) = 5 - 5i$

1

3. $(-5+3i) - (-8+2i)$

Distribute the negative sign and combine terms:

$-5+3i + 8 - 2i = (-5+8) + (3i - 2i) = 3 + i$

1

4. $(4-3i)(-5+4i)$

Use the FOIL method to multiply:

$4(-5) + 4(4i) -3i(-5) -3i(4i) = -20 + 16i + 15i - 12i^2$

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

$-20 + 31i - 12(-1) = -20 + 31i + 12 = -8 + 31i$

1

5. $(2-i)(-3+6i)$

Use the FOIL method to multiply:

$2(-3) + 2(6i) -i(-3) -i(6i) = -6 + 12i + 3i - 6i^2$

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

$-6 + 15i - 6(-1) = -6 + 15i + 6 = 15i$

1

6. $(5-3i)(5+3i)$

This is a difference of squares: $(a-b)(a+b) = a^2 - b^2$

$5^2 - (3i)^2 = 25 - 9i^2 = 25 - 9(-1) = 25 + 9 = 34$

1

7. $(-1+3i)^2$

Expand the square: $(-1+3i)(-1+3i)$

$(-1)(-1) + (-1)(3i) + (3i)(-1) + (3i)(3i) = 1 - 3i - 3i + 9i^2$

$1 - 6i + 9(-1) = 1 - 6i - 9 = -8 - 6i$

1

8. $(4-i)^2$

Expand the square: $(4-i)(4-i)$

$4(4) + 4(-i) -i(4) -i(-i) = 16 - 4i - 4i + i^2$

$16 - 8i + (-1) = 15 - 8i$

1

9. $(-2i)(5i)(-i)$

Multiply the terms:

$(-2i)(5i)(-i) = -10i^2(-i) = -10(-1)(-i) = 10(-i) = -10i$

2

0. $(6 - \sqrt{-16}) + (-4 + \sqrt{-25})$

Simplify the square roots: $\sqrt{-16} = 4i$ and $\sqrt{-25} = 5i$

$(6 - 4i) + (-4 + 5i) = (6 - 4) + (-4i + 5i) = 2 + i$

2

1. $(-2 + \sqrt{-9}) + (-1 - \sqrt{-36})$

Simplify the square roots: $\sqrt{-9} = 3i$ and $\sqrt{-36} = 6i$

$(-2 + 3i) + (-1 - 6i) = (-2 - 1) + (3i - 6i) = -3 - 3i$

3. Final Answer

1

0. $3 + i$

1

1. $0$

1

2. $5 - 5i$

1

3. $3 + i$

1

4. $-8 + 31i$

1

5. $15i$

1

6. $34$

1

7. $-8 - 6i$

1

8. $15 - 8i$

1

9. $-10i$

2

0. $2 + i$

2

1. $-3 - 3i$