Solve frac{28texttt{i}}{4+28texttt{i}}*10

$$\frac{ 28 \texttt{i} }{ 4+28 \texttt{i} } \times 10$$

Evaluate

$$\frac{49}{5}+\frac{7}{5}i=9.8+1.4i$$

Solution Steps

Multiply both numerator and denominator of \frac{28i}{4+28i} by the complex conjugate of the denominator, 4-28i.

$$\frac{28i\left(4-28i\right)}{\left(4+28i\right)\left(4-28i\right)}\times 10$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{28i\left(4-28i\right)}{4^{2}-28^{2}i^{2}}\times 10$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{28i\left(4-28i\right)}{800}\times 10$$

Multiply 28i times 4-28i.

$$\frac{28i\times 4+28\left(-28\right)i^{2}}{800}\times 10$$

By definition, i^{2} is -1.

$$\frac{28i\times 4+28\left(-28\right)\left(-1\right)}{800}\times 10$$

Do the multiplications in 28i\times 4+28\left(-28\right)\left(-1\right). Reorder the terms.

$$\frac{784+112i}{800}\times 10$$

Divide 784+112i by 800 to get \frac{49}{50}+\frac{7}{50}i.

$$\left(\frac{49}{50}+\frac{7}{50}i\right)\times 10$$

Multiply \frac{49}{50}+\frac{7}{50}i times 10.

$$\frac{49}{50}\times 10+\frac{7}{50}i\times 10$$

Do the multiplications.

$$\frac{49}{5}+\frac{7}{5}i$$

Real Part

$$\frac{49}{5} = 9\frac{4}{5} = 9.8$$

Solution Steps

Multiply both numerator and denominator of \frac{28i}{4+28i} by the complex conjugate of the denominator, 4-28i.

$$Re(\frac{28i\left(4-28i\right)}{\left(4+28i\right)\left(4-28i\right)}\times 10)$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{28i\left(4-28i\right)}{4^{2}-28^{2}i^{2}}\times 10)$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{28i\left(4-28i\right)}{800}\times 10)$$

Multiply 28i times 4-28i.

$$Re(\frac{28i\times 4+28\left(-28\right)i^{2}}{800}\times 10)$$

By definition, i^{2} is -1.

$$Re(\frac{28i\times 4+28\left(-28\right)\left(-1\right)}{800}\times 10)$$

Do the multiplications in 28i\times 4+28\left(-28\right)\left(-1\right). Reorder the terms.

$$Re(\frac{784+112i}{800}\times 10)$$

Divide 784+112i by 800 to get \frac{49}{50}+\frac{7}{50}i.

$$Re(\left(\frac{49}{50}+\frac{7}{50}i\right)\times 10)$$

Multiply \frac{49}{50}+\frac{7}{50}i times 10.

$$Re(\frac{49}{50}\times 10+\frac{7}{50}i\times 10)$$

Do the multiplications in \frac{49}{50}\times 10+\frac{7}{50}i\times 10.

$$Re(\frac{49}{5}+\frac{7}{5}i)$$

The real part of \frac{49}{5}+\frac{7}{5}i is \frac{49}{5}.

$$\frac{49}{5}$$