Solve -1+19/2i/8-3i

$$\frac{ -1+ \frac{ 19 }{ 2 } i }{ 8-3i }$$

Evaluate

$$-\frac{1}{2}+i=-0.5+i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 8+3i.

$$\frac{\left(-1+\frac{19}{2}i\right)\left(8+3i\right)}{\left(8-3i\right)\left(8+3i\right)}$$

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

$$\frac{\left(-1+\frac{19}{2}i\right)\left(8+3i\right)}{8^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(-1+\frac{19}{2}i\right)\left(8+3i\right)}{73}$$

Multiply complex numbers -1+\frac{19}{2}i and 8+3i like you multiply binomials.

$$\frac{-8-3i+\frac{19}{2}i\times 8+\frac{19}{2}\times 3i^{2}}{73}$$

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

$$\frac{-8-3i+\frac{19}{2}i\times 8+\frac{19}{2}\times 3\left(-1\right)}{73}$$

Do the multiplications in -8-3i+\frac{19}{2}i\times 8+\frac{19}{2}\times 3\left(-1\right).

$$\frac{-8-3i+76i-\frac{57}{2}}{73}$$

Combine the real and imaginary parts in -8-3i+76i-\frac{57}{2}.

$$\frac{-8-\frac{57}{2}+\left(-3+76\right)i}{73}$$

Do the additions in -8-\frac{57}{2}+\left(-3+76\right)i.

$$\frac{-\frac{73}{2}+73i}{73}$$

Divide -\frac{73}{2}+73i by 73 to get -\frac{1}{2}+i.

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

Real Part

$$-\frac{1}{2} = -0.5$$

Solution Steps

Multiply both numerator and denominator of \frac{-1+\frac{19}{2}i}{8-3i} by the complex conjugate of the denominator, 8+3i.

$$Re(\frac{\left(-1+\frac{19}{2}i\right)\left(8+3i\right)}{\left(8-3i\right)\left(8+3i\right)})$$

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

$$Re(\frac{\left(-1+\frac{19}{2}i\right)\left(8+3i\right)}{8^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(-1+\frac{19}{2}i\right)\left(8+3i\right)}{73})$$

Multiply complex numbers -1+\frac{19}{2}i and 8+3i like you multiply binomials.

$$Re(\frac{-8-3i+\frac{19}{2}i\times 8+\frac{19}{2}\times 3i^{2}}{73})$$

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

$$Re(\frac{-8-3i+\frac{19}{2}i\times 8+\frac{19}{2}\times 3\left(-1\right)}{73})$$

Do the multiplications in -8-3i+\frac{19}{2}i\times 8+\frac{19}{2}\times 3\left(-1\right).

$$Re(\frac{-8-3i+76i-\frac{57}{2}}{73})$$

Combine the real and imaginary parts in -8-3i+76i-\frac{57}{2}.

$$Re(\frac{-8-\frac{57}{2}+\left(-3+76\right)i}{73})$$

Do the additions in -8-\frac{57}{2}+\left(-3+76\right)i.

$$Re(\frac{-\frac{73}{2}+73i}{73})$$

Divide -\frac{73}{2}+73i by 73 to get -\frac{1}{2}+i.

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

The real part of -\frac{1}{2}+i is -\frac{1}{2}.

$$-\frac{1}{2}$$