Step 1: Determine the conjugate of the denominator. ( to find the conjugate just change the sign of the imaginary part ).

In this example, the conjugate of $ \color{orangered}{ 10-5i }\, $ is $ \color{blue}{ 10+5i } $.

Step 2: Multiply both the numerator and denominator by the conjugate:

$$\begin{aligned} \frac{ 13-4i }{ 10-5i } &= \frac{ 13-4i }{ 10-5i } \cdot \frac{ \color{blue}{ 10+5i } }{ \color{blue}{ 10+5i } } = \\[1 em] &= \frac{ \left( 13-4i \right) \cdot \left( 10+5i \right) }{ \left( 10-5i \right) \cdot \left( 10+5i \right) } \end{aligned} $$

Step 3: Simplify numerator and denominator (use $\color{blue}{i^2 = -1}$)

$$ \begin{aligned} \left( 13-4i \right) \cdot \left( 10+5i \right) &= 13 \cdot 10 + 13 \cdot \left(5 \,i \right) + \left( -4 \,i \right) \cdot \left(10 \right) + \left( -4 \,i \right) \cdot \left(5 \,i \right) = \\[1 em] &= 130 + 65 \, i -40 \, i -20 \color{blue}{(-1)} = \\[1 em] &= 150+25i\end{aligned} $$ $$ \begin{aligned} \left( 10-5i \right) \cdot \left( 10+5i \right) &= 10 \cdot 10 + 10 \cdot \left(5 \,i \right) + \left( -5 \,i \right) \cdot \left(10 \right) + \left( -5 \,i \right) \cdot \left(5 \,i \right) = \\[1 em] &= 100 + 50 \, i -50 \, i -25 \color{blue}{(-1)} = \\[1 em] &= 125\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 13-4i }{ 10-5i } = \frac{ 150+25i }{ 125 } = \frac{ 150 }{ 125 } + \frac{ 25 }{ 125 } i= \frac{ 6 }{ 5 }+\frac{ 1 }{ 5 }i $$

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