Answer
$\dfrac{5}{13}+\dfrac{12}{13}i$
Work Step by Step
Multiply the numerator and the denominator by the conjugate of $5-12i$ which is $5+12i$.
$\dfrac{13}{5-12i}=\dfrac{13}{5-12i}\cdot \dfrac{5+12i}{5+12i}$
Use special formula $(a+b)(a-b)=a^2-b^2$ in the denominator.
$=\dfrac{13(5+12i)}{(5)^2-(12i)^2}$
$=\dfrac{13(5+12i)}{25-144i^2}$
Use $i^2=-1$.
$=\dfrac{13(5+12i)}{25-144(-1)}$
Simplify.
$=\dfrac{13(5+12i)}{25+144}$
$=\dfrac{13(5+12i)}{169}$
$\require{cancel}=\dfrac{\cancel{13}(5+12i)}{\cancel{169}^{13}}$
$=\dfrac{5+12i}{13}$
$=\dfrac{5}{13}+\dfrac{12}{13}i$
Hence, the solution in the standard form is $\dfrac{5}{13}+\dfrac{12}{13}i$.
