Question

If x=(10-3i) and y=(3-10i), then xy=? and x/y=?

Answer 1

Answer:

$xy=-109i$

$\frac{x}{y}=\frac{60}{109}+\frac{91}{109}i$

Step-by-step explanation:

$xy=(10-3i)(3-10i)$

To compute x times y must use foil on the right hand side.

First:  $10(3)=30$

Outer: $10(-10i)=-100i$

Inner: $-3i(3)=-9i$

Last: $-3i(-10i)=30i^2=-30$

------------------------------------Add like terms:

$-109i$

----------------

$\frac{x}{y}$

$\frac{10-3i}{3-10i}$

Multiply top and bottom by bottom's conjugate:

$\frac{(10-3i)(3+10i)}{(3-10i)(3+10i)}$

Foil the top and just do first and last of Foil for the bottom since the bottom contains multiplying conjugates:

$\frac{30+100i-9i-30i^2}{9-100i^2}$

Replace $i^2$ with -1:

$\frac{30+91i+30}{9+100}$

$\frac{60+91i}{109}$

$\frac{60}{109}+\frac{91}{109}i$