How do you simplify #3/(6-3i)#?

1 Answer
Oct 20, 2016

Answer:

#2/5+1/5i#

Explanation:

To simplify this fraction, we must multiply the numerator/denominator by the #color(blue)"complex conjugate"# of the denominator.
This ensures that the denominator is a real number.

Given a complex number #z=x±yi# then the complex conjugate is.

#color(red)(bar(ul(|color(white)(2/2)color(black)(barz=x∓yi)color(white)(2/2)|)))#

Note that the real part remains unchanged while the #color(red)"sign"# of the imaginary part is reversed.

Hence the conjugate of #6-3i" is " 6+3i#

and #(6-3i)(6+3i)=36+9=45larr" a real number"#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(i^2=(sqrt(-1))^2=-1)color(white)(2/2)|)))#

Multiply numerator/denominator by 6 + 3i

#(3(6+3i))/((6-3i)(6+3i))=(18+9i)/45=18/45+9/45i=2/5+1/5i#