Question

express as a complex number in simplest a+bi form: \\(\dfrac{23 - 11i}{-9 - 7i}\\)

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by the conjugate of the denominator $-9+7i$:
$$\frac{23-11i}{-9-7i} \times \frac{-9+7i}{-9+7i}$$

Step2: Expand numerator

Use FOIL method for numerator:
$$(23)(-9) + (23)(7i) + (-11i)(-9) + (-11i)(7i) = -207 + 161i + 99i -77i^2$$
Substitute $i^2=-1$:
$$-207 + 260i -77(-1) = -207 + 260i +77 = -130 + 260i$$

Step3: Expand denominator

Use difference of squares for denominator:
$$(-9)^2 - (7i)^2 = 81 - 49i^2$$
Substitute $i^2=-1$:
$$81 - 49(-1) = 81 + 49 = 130$$

Step4: Simplify fraction

Divide numerator terms by denominator:
$$\frac{-130}{130} + \frac{260i}{130}$$

Answer:

$-1 + 2i$