[tex] \huge \boxed{\mathfrak{Answer} \downarrow}[/tex]
[tex] \large \bf\frac { 5 - 3 i } { - 2 - 9 i } \\ [/tex]
Multiply both numerator and denominator of [tex]\sf \frac{5-3i}{-2-9i} \\ [/tex] by the complex conjugate of the denominator, -2+9i.
[tex] \large \bf \: Re(\frac{\left(5-3i\right)\left(-2+9i\right)}{\left(-2-9i\right)\left(-2+9i\right)}) \\ [/tex]
Multiplication can be transformed into difference of squares using the rule: [tex]\sf\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}[/tex].
[tex] \large \bf \: Re(\frac{\left(5-3i\right)\left(-2+9i\right)}{\left(-2\right)^{2}-9^{2}i^{2}}) \\ [/tex]
By definition, i² is -1. Calculate the denominator.
[tex] \large \bf \: Re(\frac{\left(5-3i\right)\left(-2+9i\right)}{85}) \\ [/tex]
Multiply complex numbers 5-3i and -2+9i in the same way as you multiply binomials.
[tex] \large \bf \: Re(\frac{5\left(-2\right)+5\times \left(9i\right)-3i\left(-2\right)-3\times 9i^{2}}{85}) \\ [/tex]
Do the multiplications in [tex]\sf5\left(-2\right)+5\times \left(9i\right)-3i\left(-2\right)-3\times 9\left(-1\right)[/tex].
[tex] \large \bf \: Re(\frac{-10+45i+6i+27}{85}) \\ [/tex]
Combine the real and imaginary parts in -10+45i+6i+27.
[tex] \large \bf \: Re(\frac{-10+27+\left(45+6\right)i}{85}) \\ [/tex]
Do the additions in [tex]\sf-10+27+\left(45+6\right)i[/tex].
[tex] \large \bf Re(\frac{17+51i}{85}) \\ [/tex]
Divide 17+51i by 85 to get [tex]\sf\frac{1}{5}+\frac{3}{5}i \\ [/tex].
[tex] \large \bf \: Re(\frac{1}{5}+\frac{3}{5}i) \\ [/tex]
The real part of [tex]\sf \frac{1}{5}+\frac{3}{5}i \\ [/tex] is [tex]\sf \frac{1}{5} \\ [/tex].
[tex] \large \boxed{\bf\frac{1}{5} = 0.2} \\ [/tex]
