Solve -8+24i/209+96i

$$\frac { - 8 + 24 i } { 209 + 96 i }$$

Evaluate

$$\frac{632}{52897}+\frac{5784}{52897}i\approx 0.011947748+0.109344575i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 209-96i.

$$\frac{\left(-8+24i\right)\left(209-96i\right)}{\left(209+96i\right)\left(209-96i\right)}$$

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

$$\frac{\left(-8+24i\right)\left(209-96i\right)}{209^{2}-96^{2}i^{2}}$$

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

$$\frac{\left(-8+24i\right)\left(209-96i\right)}{52897}$$

Multiply complex numbers -8+24i and 209-96i like you multiply binomials.

$$\frac{-8\times 209-8\times \left(-96i\right)+24i\times 209+24\left(-96\right)i^{2}}{52897}$$

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

$$\frac{-8\times 209-8\times \left(-96i\right)+24i\times 209+24\left(-96\right)\left(-1\right)}{52897}$$

Do the multiplications in -8\times 209-8\times \left(-96i\right)+24i\times 209+24\left(-96\right)\left(-1\right).

$$\frac{-1672+768i+5016i+2304}{52897}$$

Combine the real and imaginary parts in -1672+768i+5016i+2304.

$$\frac{-1672+2304+\left(768+5016\right)i}{52897}$$

Do the additions in -1672+2304+\left(768+5016\right)i.

$$\frac{632+5784i}{52897}$$

Divide 632+5784i by 52897 to get \frac{632}{52897}+\frac{5784}{52897}i.

$$\frac{632}{52897}+\frac{5784}{52897}i$$

Real Part

$$\frac{632}{52897} = 0.011947747509310547$$

Solution Steps

Multiply both numerator and denominator of \frac{-8+24i}{209+96i} by the complex conjugate of the denominator, 209-96i.

$$Re(\frac{\left(-8+24i\right)\left(209-96i\right)}{\left(209+96i\right)\left(209-96i\right)})$$

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

$$Re(\frac{\left(-8+24i\right)\left(209-96i\right)}{209^{2}-96^{2}i^{2}})$$

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

$$Re(\frac{\left(-8+24i\right)\left(209-96i\right)}{52897})$$

Multiply complex numbers -8+24i and 209-96i like you multiply binomials.

$$Re(\frac{-8\times 209-8\times \left(-96i\right)+24i\times 209+24\left(-96\right)i^{2}}{52897})$$

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

$$Re(\frac{-8\times 209-8\times \left(-96i\right)+24i\times 209+24\left(-96\right)\left(-1\right)}{52897})$$

Do the multiplications in -8\times 209-8\times \left(-96i\right)+24i\times 209+24\left(-96\right)\left(-1\right).

$$Re(\frac{-1672+768i+5016i+2304}{52897})$$

Combine the real and imaginary parts in -1672+768i+5016i+2304.

$$Re(\frac{-1672+2304+\left(768+5016\right)i}{52897})$$

Do the additions in -1672+2304+\left(768+5016\right)i.

$$Re(\frac{632+5784i}{52897})$$

Divide 632+5784i by 52897 to get \frac{632}{52897}+\frac{5784}{52897}i.

$$Re(\frac{632}{52897}+\frac{5784}{52897}i)$$

The real part of \frac{632}{52897}+\frac{5784}{52897}i is \frac{632}{52897}.

$$\frac{632}{52897}$$