Solve frac{13+12texttt{i}}{(13-12texttt{i})*(13+12texttt{i})}

$$\frac{ 13+12 \texttt{i} }{ (13-12 \texttt{i} ) \times (13+12 \texttt{i} ) }$$

Evaluate

$$\frac{13}{313}+\frac{12}{313}i\approx 0.041533546+0.038338658i$$

Solution Steps

Multiply complex numbers 13-12i and 13+12i like you multiply binomials.

$$\frac{13+12i}{13\times 13+13\times \left(12i\right)-12i\times 13-12\times 12i^{2}}$$

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

$$\frac{13+12i}{13\times 13+13\times \left(12i\right)-12i\times 13-12\times 12\left(-1\right)}$$

Do the multiplications in 13\times 13+13\times \left(12i\right)-12i\times 13-12\times 12\left(-1\right).

$$\frac{13+12i}{169+156i-156i+144}$$

Combine the real and imaginary parts in 169+156i-156i+144.

$$\frac{13+12i}{169+144+\left(156-156\right)i}$$

Do the additions in 169+144+\left(156-156\right)i.

$$\frac{13+12i}{313}$$

Divide 13+12i by 313 to get \frac{13}{313}+\frac{12}{313}i.

$$\frac{13}{313}+\frac{12}{313}i$$

Real Part

$$\frac{13}{313} = 0.04153354632587859$$

Solution Steps

Multiply complex numbers 13-12i and 13+12i like you multiply binomials.

$$Re(\frac{13+12i}{13\times 13+13\times \left(12i\right)-12i\times 13-12\times 12i^{2}})$$

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

$$Re(\frac{13+12i}{13\times 13+13\times \left(12i\right)-12i\times 13-12\times 12\left(-1\right)})$$

Do the multiplications in 13\times 13+13\times \left(12i\right)-12i\times 13-12\times 12\left(-1\right).

$$Re(\frac{13+12i}{169+156i-156i+144})$$

Combine the real and imaginary parts in 169+156i-156i+144.

$$Re(\frac{13+12i}{169+144+\left(156-156\right)i})$$

Do the additions in 169+144+\left(156-156\right)i.

$$Re(\frac{13+12i}{313})$$

Divide 13+12i by 313 to get \frac{13}{313}+\frac{12}{313}i.

$$Re(\frac{13}{313}+\frac{12}{313}i)$$

The real part of \frac{13}{313}+\frac{12}{313}i is \frac{13}{313}.

$$\frac{13}{313}$$