$$\dfrac{1+i\dfrac{26-7i}{25}-2\dfrac{24+7i}{25}}{1+i}=\dfrac{-2+14i}{25}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1+i\frac{26-7i}{25}-2\frac{24+7i}{25}}{1+i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1+\frac{-7i^2+26i}{25}-\frac{14i+48}{25}}{1+i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1+\frac{7+26i}{25}-\frac{14i+48}{25}}{1+i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{\frac{26i+32}{25}-\frac{14i+48}{25}}{1+i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{\frac{12i-16}{25}}{1+i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{12i-16}{25i+25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{-2+14i}{25}\end{aligned} $$ | |
| ① | Multiply $i$ by $ \dfrac{26-7i}{25} $ to get $ \dfrac{-7i^2+26i}{25} $. Step 1: Write $ i $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} i \cdot \frac{26-7i}{25} & \xlongequal{\text{Step 1}} \frac{i}{\color{red}{1}} \cdot \frac{26-7i}{25} \xlongequal{\text{Step 2}} \frac{ i \cdot \left( 26-7i \right) }{ 1 \cdot 25 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 26i-7i^2 }{ 25 } = \frac{-7i^2+26i}{25} \end{aligned} $$ |
| ② | Multiply $2$ by $ \dfrac{24+7i}{25} $ to get $ \dfrac{14i+48}{25} $. Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 2 \cdot \frac{24+7i}{25} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{24+7i}{25} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 24+7i \right) }{ 1 \cdot 25 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 48+14i }{ 25 } = \frac{14i+48}{25} \end{aligned} $$ |
| ③ | $$ -7i^2 = -7 \cdot (-1) = 7 $$ |
| ④ | Multiply $2$ by $ \dfrac{24+7i}{25} $ to get $ \dfrac{14i+48}{25} $. Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 2 \cdot \frac{24+7i}{25} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{24+7i}{25} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 24+7i \right) }{ 1 \cdot 25 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 48+14i }{ 25 } = \frac{14i+48}{25} \end{aligned} $$ |
| ⑤ | Add $1$ and $ \dfrac{7+26i}{25} $ to get $ \dfrac{ \color{purple}{ 26i+32 } }{ 25 }$. Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ⑥ | Multiply $2$ by $ \dfrac{24+7i}{25} $ to get $ \dfrac{14i+48}{25} $. Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 2 \cdot \frac{24+7i}{25} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{24+7i}{25} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 24+7i \right) }{ 1 \cdot 25 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 48+14i }{ 25 } = \frac{14i+48}{25} \end{aligned} $$ |
| ⑦ | Subtract $ \dfrac{14i+48}{25} $ from $ \dfrac{26i+32}{25} $ to get $ \dfrac{12i-16}{25} $. To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator. $$ \begin{aligned} \frac{26i+32}{25} - \frac{14i+48}{25} & = \frac{26i+32}{\color{blue}{25}} - \frac{14i+48}{\color{blue}{25}} = \\[1ex] &=\frac{ 26i+32 - \left( 14i+48 \right) }{ \color{blue}{ 25 }}= \frac{12i-16}{25} \end{aligned} $$ |
| ⑧ | Divide $ \dfrac{12i-16}{25} $ by $ 1+i $ to get $ \dfrac{12i-16}{25i+25} $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{12i-16}{25} }{1+i} & \xlongequal{\text{Step 1}} \frac{12i-16}{25} \cdot \frac{\color{blue}{1}}{\color{blue}{1+i}} \xlongequal{\text{Step 2}} \frac{ \left( 12i-16 \right) \cdot 1 }{ 25 \cdot \left( 1+i \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12i-16 }{ 25+25i } = \frac{12i-16}{25i+25} \end{aligned} $$ |
| ⑨ | Divide $ \, -16+12i \, $ by $ \, 25+25i \, $ to get $\,\, \dfrac{-2+14i}{25} $. ( view steps ) |