Question

Which of the following is the quotient of the complex numbers below ?

(10+2i)/(5-6i)

Answer 1

Answer:

$\frac{38}{61} + \frac{70}{61}i$

Step-by-step explanation:

Given: $\frac{10 + 2i}{5 - 6i}$

We multiply both the numerator and denominator by the complex conjugate of the denominator.

We have:

$\frac{10 + 2i}{5 - 6i} = \frac{10 + 2i}{5 - 6i} \times \frac{5 + 6i}{5 + 6i}$

This equals $\frac{(10 - 2i)(5 - 6i)}{(5 - 6i)(5 + 6i)}$

Note that the denominator is of the form, $(a + ib)(a - ib)$.

This is equal to $a^2 + b^2$.

Multiplying the numerator term - wise and applying the above formula for denominator, we have:

$\frac{(10 + 2i)(5 + 6i)}{5^2 + 6^2}$

$\frac{50 + 60i + 10i - 12}{61}$

$= \frac{38 + 70i}{61}$

$\frac{38}{61} + \frac{70}{61}i$ is the required answer.