Let z 5 i and w 5 6i Compute the following and express yo

Let

z = 5 + i

and

w = 5 6i.

Compute the following and express your answer in

a + bi

form.

Solution

z = 5 + i , w = 5 - 6i

w/z = (5 + i)/(5 - 6i)

Rationalize denominator i.e, Multiply and divide by (5 + 6i)

==> [(5 + i)/(5 - 6i)] [(5 + 6i)/(5 + 6i)]

==> [(5 + i)(5 + 6i)]/[ (5² - (6i)²]           ; since (a - b)(a + b) = a² - b²

==> [25 + 30i + 5i + 6i²] / [25 - 36i²]        ; i² = -1

==> [25 + 35i + 6(-1)] / [25 - 36(-1)]

==> (19 + 35i)/(25 + 36)

==> (19 + 35i)/61

==> (19/61) + (35/61)i

Hence w/z = (19/61) + (35/61)i