Let z 44 5i and w 2 7i Compute the following and express

Let

z = 44 + 5i

and

w = 2 7i.

Compute the following and express your answer in

a + bi

form.

z/w

Solution

z = 44 +5i , w = 2 - 7i

z/w = (44 +5i)/(2 - 7i)

Rationalize denominator i.e, Multiply and divide by (2 + 7i)

==> [(44 +5i)/(2 - 7i)] [(2 + 7i)/(2 + 7i)]

==> [(44 +5i)(2 + 7i)]/[ (2² - (7i)²]           ; since (a - b)(a + b) = a² - b²

==> [88 + 308i + 10i + 35i²] / [4 - 49i²]        ; i² = -1

==> [88 + 318i + 35(-1)] / [4 - 49(-1)]

==> (53 + 318i)/(4 + 49)

==> (53 + 318i)/53

==> (53/53) + (318/53)i

==> 1 + 6i

Hence z/w = 1 + 6i