Solve 18i/2-3i+1-i/-2+3i

$$\frac { 18 i } { 2 - 3 i } + \frac { 1 - i } { - 2 + 3 i }$$

Evaluate

$$-\frac{59}{13}+\frac{35}{13}i\approx -4.538461538+2.692307692i$$

Solution Steps

Multiply both numerator and denominator of \frac{18i}{2-3i} by the complex conjugate of the denominator, 2+3i.

$$\frac{18i\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}+\frac{1-i}{-2+3i}$$

Do the multiplications in \frac{18i\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}.

$$\frac{-54+36i}{13}+\frac{1-i}{-2+3i}$$

Divide -54+36i by 13 to get -\frac{54}{13}+\frac{36}{13}i.

$$-\frac{54}{13}+\frac{36}{13}i+\frac{1-i}{-2+3i}$$

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

$$-\frac{54}{13}+\frac{36}{13}i+\frac{\left(1-i\right)\left(-2-3i\right)}{\left(-2+3i\right)\left(-2-3i\right)}$$

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

$$-\frac{54}{13}+\frac{36}{13}i+\frac{-5-i}{13}$$

Divide -5-i by 13 to get -\frac{5}{13}-\frac{1}{13}i.

$$-\frac{54}{13}+\frac{36}{13}i+\left(-\frac{5}{13}-\frac{1}{13}i\right)$$

Add -\frac{54}{13}+\frac{36}{13}i and -\frac{5}{13}-\frac{1}{13}i to get -\frac{59}{13}+\frac{35}{13}i.

$$-\frac{59}{13}+\frac{35}{13}i$$

Real Part

$$-\frac{59}{13} = -4\frac{7}{13} = -4.538461538461538$$

Solution Steps

Multiply both numerator and denominator of \frac{18i}{2-3i} by the complex conjugate of the denominator, 2+3i.

$$Re(\frac{18i\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}+\frac{1-i}{-2+3i})$$

Do the multiplications in \frac{18i\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}.

$$Re(\frac{-54+36i}{13}+\frac{1-i}{-2+3i})$$

Divide -54+36i by 13 to get -\frac{54}{13}+\frac{36}{13}i.

$$Re(-\frac{54}{13}+\frac{36}{13}i+\frac{1-i}{-2+3i})$$

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

$$Re(-\frac{54}{13}+\frac{36}{13}i+\frac{\left(1-i\right)\left(-2-3i\right)}{\left(-2+3i\right)\left(-2-3i\right)})$$

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

$$Re(-\frac{54}{13}+\frac{36}{13}i+\frac{-5-i}{13})$$

Divide -5-i by 13 to get -\frac{5}{13}-\frac{1}{13}i.

$$Re(-\frac{54}{13}+\frac{36}{13}i+\left(-\frac{5}{13}-\frac{1}{13}i\right))$$

Add -\frac{54}{13}+\frac{36}{13}i and -\frac{5}{13}-\frac{1}{13}i to get -\frac{59}{13}+\frac{35}{13}i.

$$Re(-\frac{59}{13}+\frac{35}{13}i)$$

The real part of -\frac{59}{13}+\frac{35}{13}i is -\frac{59}{13}.

$$-\frac{59}{13}$$