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Question

Answer

$$(\dfrac{80}{3}-52.053i)(\dfrac{80}{3}+52.053i)=\dfrac{30736}{9}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}(\frac{80}{3}-52.053i)(\frac{80}{3}+52.053i)& \xlongequal{ }(\frac{80}{3}-52i)(\frac{80}{3}+52i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-156i+80}{3}\frac{156i+80}{3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-24336i^2+6400}{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{24336+6400}{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{30736}{9}\end{aligned} $$
①

Subtract $52i$ from $ \dfrac{80}{3} $ to get $ \dfrac{ \color{purple}{ -156i+80 } }{ 3 }$.

Step 1: Write $ 52i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{80}{3} -52i & \xlongequal{\text{Step 1}} \frac{80}{3} - \frac{52i}{\color{red}{1}} = \frac{ 80 }{ 3 } - \frac{ 52i \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 80 } }{ 3 } - \frac{ \color{purple}{ 156i } }{ 3 }=\frac{ \color{purple}{ -156i+80 } }{ 3 } \end{aligned} $$
②

Add $ \dfrac{80}{3} $ and $ 52i $ to get $ \dfrac{ \color{purple}{ 156i+80 } }{ 3 }$.

Step 1: Write $ 52i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{80}{3} +52i & \xlongequal{\text{Step 1}} \frac{80}{3} + \frac{52i}{\color{red}{1}} = \frac{ 80 }{ 3 } + \frac{ 52i \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 80 } }{ 3 } + \frac{ \color{purple}{ 156i } }{ 3 }=\frac{ \color{purple}{ 156i+80 } }{ 3 } \end{aligned} $$
③

Multiply $ \dfrac{-156i+80}{3} $ by $ \dfrac{156i+80}{3} $ to get $ \dfrac{-24336i^2+6400}{9} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-156i+80}{3} \cdot \frac{156i+80}{3} & \xlongequal{\text{Step 1}} \frac{ \left( -156i+80 \right) \cdot \left( 156i+80 \right) }{ 3 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ -24336i^2 -\cancel{12480i}+ \cancel{12480i}+6400 }{ 9 } = \frac{-24336i^2+6400}{9} \end{aligned} $$
④
$$ -24336i^2 = -24336 \cdot (-1) = 24336 $$
⑤

Simplify numerator

$$ \color{blue}{24336} + \color{blue}{6400} = \color{blue}{30736} $$
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