Question 420562
I'm a little confused by the 8? I'm assuming that it is a typo.<br>
The multiplicative inverse of 3+5i is what you misgh expect it to be:
{{{1/(3+5i)}}}
However this is not the proper form. We want something in the standard form:
a + bi
where a and b are real numbers.<br>
To put {{{1/(3+5i)}} in standard form we multiply its numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3+5i is 3-5i. So we multiply the numerator and denominator by 3-5i:
{{{(1/(3+5i))((3-5i)/3-5i))}}}
Multiplying the numerator is simple. To multiply the denominator we can use FOIL or the {{{(a+b)(a-b) = a^2-b^2}}} pattern. I prefer using the pattern:
{{{(3-5i)/((3)^2-(5i)^2)}}}
which simplifies as follows:
{{{(3-5i)/(9-25i^2)}}}
{{{(3-5i)/(9-25(-1))}}}  (since {{{i^2 = -1}}})
{{{(3-5i)/(9+25)}}}
{{{(3-5i)/34}}}
For standard form we need to split up this fraction:
{{{3/34+ (-5i)/34}}}
or
{{{3/34+ ((-5)/34)i}}}