Question

divide.
\\(\frac{-5}{-6 + 3i}\\)
write your answer as a complex number in standard form

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $-6-3i$ (the complex conjugate of $-6+3i$) to eliminate the imaginary part in the denominator.
$$\frac{-5}{-6+3i} \times \frac{-6-3i}{-6-3i} = \frac{-5(-6-3i)}{(-6+3i)(-6-3i)}$$

Step2: Expand numerator and denominator

Calculate the products for numerator and denominator. Use the difference of squares for the denominator: $(a+b)(a-b)=a^2-b^2$.
Numerator: $-5(-6-3i) = 30 + 15i$
Denominator: $(-6)^2 - (3i)^2 = 36 - 9i^2$

Step3: Simplify using $i^2=-1$

Substitute $i^2=-1$ into the denominator and simplify.
Denominator: $36 - 9(-1) = 36 + 9 = 45$
So the expression becomes $\frac{30 + 15i}{45}$

Step4: Split and simplify fractions

Separate the fraction into real and imaginary parts, then reduce each fraction.
$$\frac{30}{45} + \frac{15}{45}i = \frac{2}{3} + \frac{1}{3}i$$

Answer:

$\frac{2}{3} + \frac{1}{3}i$