Multiply $7500$ by $ \dfrac{i}{13} $ to get $ \dfrac{ 7500i }{ 13 } $.

Step 1: Write $ 7500 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 7500 \cdot \frac{i}{13} & \xlongequal{\text{Step 1}} \frac{7500}{\color{red}{1}} \cdot \frac{i}{13} \xlongequal{\text{Step 2}} \frac{ 7500 \cdot i }{ 1 \cdot 13 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 7500i }{ 13 } \end{aligned} $$

Add $ \dfrac{1500}{13} $ and $ \dfrac{7500i}{13} $ to get $ \dfrac{7500i+1500}{13} $.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{1500}{13} + \frac{7500i}{13} & = \frac{1500}{\color{blue}{13}} + \frac{7500i}{\color{blue}{13}} =\frac{ 1500 + 7500i }{ \color{blue}{ 13 }} = \\[1ex] &= \frac{7500i+1500}{13} \end{aligned} $$

Divide $1$ by $ \dfrac{7500i+1500}{13} $ to get $ \dfrac{ 13 }{ 7500i+1500 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{7500i+1500}}{\color{blue}{13}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{13}}{\color{blue}{7500i+1500}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{13}{7500i+1500} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 13 }{ 1 \cdot \left( 7500i+1500 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 13 }{ 7500i+1500 } \end{aligned} $$

Add $ \dfrac{13}{7500i+1500} $ and $ \dfrac{1}{93+42i} $ to get $ \dfrac{ \color{purple}{ 2682i+903 } }{ 105000i^2+253500i+46500 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 14i+31 }$ and the second by $\color{blue}{ 2500i+500 }$.

$$ \begin{aligned} \frac{13}{7500i+1500} + \frac{1}{93+42i} & = \frac{ 13 \cdot \color{blue}{ \left( 14i+31 \right) }}{ \left( 7500i+1500 \right) \cdot \color{blue}{ \left( 14i+31 \right) }} + \frac{ 1 \cdot \color{blue}{ \left( 2500i+500 \right) }}{ \left( 93+42i \right) \cdot \color{blue}{ \left( 2500i+500 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 182i+403 } }{ 105000i^2+232500i+21000i+46500 } + \frac{ \color{purple}{ 2500i+500 } }{ 105000i^2+232500i+21000i+46500 } = \\[1ex] &=\frac{ \color{purple}{ 2682i+903 } }{ 105000i^2+253500i+46500 } \end{aligned} $$

$$ 105000i^2 = 105000 \cdot (-1) = -105000 $$

Combine like terms:

$$ \color{blue}{-105000} +253500i+ \color{blue}{46500} = 253500i \color{blue}{-58500} $$

Divide $ \, 903+2682i \, $ by $ \, -58500+253500i \, $ to get $\,\, \dfrac{10719-6595i}{1157000} $.
( view steps )

Divide $1$ by $ \dfrac{10719-6595i}{1157000} $ to get $ \dfrac{1157000}{-6595i+10719} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{10719-6595i}}{\color{blue}{1157000}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{1157000}}{\color{blue}{10719-6595i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{1157000}{10719-6595i} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 1157000 }{ 1 \cdot \left( 10719-6595i \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 1157000 }{ 10719-6595i } = \frac{1157000}{-6595i+10719} \end{aligned} $$