Solve 1/5+10texttt{i}+1/3-4texttt{i}=

$$\frac{ 1 }{ 5+10 \texttt{i} } + \frac{ 1 }{ 3-4 \texttt{i} } =$$

Evaluate

$$\frac{4}{25}+\frac{2}{25}i=0.16+0.08i$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{5+10i} by the complex conjugate of the denominator, 5-10i.

$$\frac{1\left(5-10i\right)}{\left(5+10i\right)\left(5-10i\right)}+\frac{1}{3-4i}$$

Do the multiplications in \frac{1\left(5-10i\right)}{\left(5+10i\right)\left(5-10i\right)}.

$$\frac{5-10i}{125}+\frac{1}{3-4i}$$

Divide 5-10i by 125 to get \frac{1}{25}-\frac{2}{25}i.

$$\frac{1}{25}-\frac{2}{25}i+\frac{1}{3-4i}$$

Multiply both numerator and denominator of \frac{1}{3-4i} by the complex conjugate of the denominator, 3+4i.

$$\frac{1}{25}-\frac{2}{25}i+\frac{1\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}$$

Do the multiplications in \frac{1\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}.

$$\frac{1}{25}-\frac{2}{25}i+\frac{3+4i}{25}$$

Divide 3+4i by 25 to get \frac{3}{25}+\frac{4}{25}i.

$$\frac{1}{25}-\frac{2}{25}i+\left(\frac{3}{25}+\frac{4}{25}i\right)$$

Add \frac{1}{25}-\frac{2}{25}i and \frac{3}{25}+\frac{4}{25}i to get \frac{4}{25}+\frac{2}{25}i.

$$\frac{4}{25}+\frac{2}{25}i$$

Real Part

$$\frac{4}{25} = 0.16$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{5+10i} by the complex conjugate of the denominator, 5-10i.

$$Re(\frac{1\left(5-10i\right)}{\left(5+10i\right)\left(5-10i\right)}+\frac{1}{3-4i})$$

Do the multiplications in \frac{1\left(5-10i\right)}{\left(5+10i\right)\left(5-10i\right)}.

$$Re(\frac{5-10i}{125}+\frac{1}{3-4i})$$

Divide 5-10i by 125 to get \frac{1}{25}-\frac{2}{25}i.

$$Re(\frac{1}{25}-\frac{2}{25}i+\frac{1}{3-4i})$$

Multiply both numerator and denominator of \frac{1}{3-4i} by the complex conjugate of the denominator, 3+4i.

$$Re(\frac{1}{25}-\frac{2}{25}i+\frac{1\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)})$$

Do the multiplications in \frac{1\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}.

$$Re(\frac{1}{25}-\frac{2}{25}i+\frac{3+4i}{25})$$

Divide 3+4i by 25 to get \frac{3}{25}+\frac{4}{25}i.

$$Re(\frac{1}{25}-\frac{2}{25}i+\left(\frac{3}{25}+\frac{4}{25}i\right))$$

Add \frac{1}{25}-\frac{2}{25}i and \frac{3}{25}+\frac{4}{25}i to get \frac{4}{25}+\frac{2}{25}i.

$$Re(\frac{4}{25}+\frac{2}{25}i)$$

The real part of \frac{4}{25}+\frac{2}{25}i is \frac{4}{25}.

$$\frac{4}{25}$$