Question 1206092
<font color=black size=3>
Answer:
{{{z[3] = 35/26+expr(45/26)i}}}



Work Shown


{{{1/(z[3]) = 1/(z[1])+1/(z[2])}}}


{{{1/(z[3]) = (z[2])/(z[1]z[2])+(z[1])/(z[1]z[2])}}} Get every fraction on the right hand side in terms of the LCD


{{{1/(z[3]) = (z[2]+z[1])/(z[1]z[2])}}}


{{{z[3] = (z[1]z[2])/(z[2]+z[1])}}} Apply reciprocal to both sides.


{{{z[3] = ((2+i)(-3+4i))/(-3+4i+2+i)}}}


{{{z[3] = (-10+5i)/(-1+5i)}}}  See scratch work section shown below.


{{{z[3] = ((-10+5i)(-1-5i))/((-1+5i)(-1-5i))}}} Multiply top and bottom by conjugate of the denominator so the denominator becomes a real number.


{{{z[3] = (35+45i)/(26)}}} Follow similar steps as the scratch work section shown below. Note: (a+bi)(a-bi) = a^2+b^2


{{{z[3] = 35/26+expr(45/26)i}}}


Complex number {{{z[3]}}} is of the form a+bi where {{{a = 35/26}}} and {{{b = 45/26}}}


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


Scratch Work


{{{(2+i)(-3+4i) = 2(-3+4i)+i(-3+4i)}}}


{{{(2+i)(-3+4i) = -6+8i-3i+4i^2}}}


{{{(2+i)(-3+4i) = -6+8i-3i+4(-1)}}}


{{{(2+i)(-3+4i) = -6+8i-3i-4}}}


{{{(2+i)(-3+4i) = -10+5i}}}
Another approach for this scratch work section is to use the FOIL rule or the box method.
</font>