Question

divide.
\\(\frac{4i}{-1 + 6i}\\)
write your answer as a complex number in standard form.

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The denominator is $-1 + 6i$, so its conjugate is $-1 - 6i$. We multiply the fraction $\frac{4i}{-1 + 6i}$ by $\frac{-1 - 6i}{-1 - 6i}$:
$$\frac{4i(-1 - 6i)}{(-1 + 6i)(-1 - 6i)}$$

Step2: Expand the numerator and the denominator.

First, expand the numerator: $4i(-1 - 6i)= -4i - 24i^2$. Since $i^2 = -1$, this becomes $-4i - 24(-1)=24 - 4i$.
Next, expand the denominator using the difference of squares formula $(a + b)(a - b)=a^2 - b^2$. Here, $a = -1$ and $b = 6i$, so $(-1)^2 - (6i)^2 = 1 - 36i^2$. Substituting $i^2 = -1$, we get $1 - 36(-1)=1 + 36 = 37$.

Step3: Write the result as a complex number.

Now we have $\frac{24 - 4i}{37}$, which can be split into $\frac{24}{37} - \frac{4}{37}i$.

Answer:

$\frac{24}{37} - \frac{4}{37}i$