Question

1. \\(\frac{-2 + i}{-1 - 8i}\\)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $-1+8i$ (the conjugate of $-1-8i$)
$$\frac{-2+i}{-1-8i} \times \frac{-1+8i}{-1+8i} = \frac{(-2+i)(-1+8i)}{(-1-8i)(-1+8i)}$$

Step2: Expand numerator

Use FOIL method for numerator
$$(-2)(-1) + (-2)(8i) + i(-1) + i(8i) = 2 -16i -i +8i^2$$
Since $i^2=-1$, substitute:
$$2 -17i +8(-1) = 2 -17i -8 = -6 -17i$$

Step3: Expand denominator

Use difference of squares: $(a-b)(a+b)=a^2-b^2$
$$(-1)^2 - (8i)^2 = 1 -64i^2$$
Substitute $i^2=-1$:
$$1 -64(-1) = 1 +64 = 65$$

Step4: Simplify the fraction

Divide numerator by denominator
$$\frac{-6 -17i}{65} = -\frac{6}{65} - \frac{17}{65}i$$

Answer:

$-\frac{6}{65} - \frac{17}{65}i$