Solve (1-4i)/(2-3i)(2+3i/2+3i)

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

Evaluate

$$\frac{14}{13}-\frac{5}{13}i\approx 1.076923077-0.384615385i$$

Solution Steps

Divide 2+3i by 2+3i to get 1.

$$\frac{1-4i}{2-3i}\times 1$$

Multiply both numerator and denominator of \frac{1-4i}{2-3i} by the complex conjugate of the denominator, 2+3i.

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

Do the multiplications in \frac{\left(1-4i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}.

$$\frac{14-5i}{13}\times 1$$

Divide 14-5i by 13 to get \frac{14}{13}-\frac{5}{13}i.

$$\left(\frac{14}{13}-\frac{5}{13}i\right)\times 1$$

Multiply \frac{14}{13}-\frac{5}{13}i and 1 to get \frac{14}{13}-\frac{5}{13}i.

$$\frac{14}{13}-\frac{5}{13}i$$

Real Part

$$\frac{14}{13} = 1\frac{1}{13} = 1.0769230769230769$$

Solution Steps

Divide 2+3i by 2+3i to get 1.

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

Multiply both numerator and denominator of \frac{1-4i}{2-3i} by the complex conjugate of the denominator, 2+3i.

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

Do the multiplications in \frac{\left(1-4i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}.

$$Re(\frac{14-5i}{13}\times 1)$$

Divide 14-5i by 13 to get \frac{14}{13}-\frac{5}{13}i.

$$Re(\left(\frac{14}{13}-\frac{5}{13}i\right)\times 1)$$

Multiply \frac{14}{13}-\frac{5}{13}i and 1 to get \frac{14}{13}-\frac{5}{13}i.

$$Re(\frac{14}{13}-\frac{5}{13}i)$$

The real part of \frac{14}{13}-\frac{5}{13}i is \frac{14}{13}.

$$\frac{14}{13}$$