Solve 9+28i/92+8i

$$\frac { 9 + 28 i } { 92 + 8 i }$$

Evaluate

$$\frac{263}{2132}+\frac{313}{1066}i\approx 0.123358349+0.293621013i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 92-8i.

$$\frac{\left(9+28i\right)\left(92-8i\right)}{\left(92+8i\right)\left(92-8i\right)}$$

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

$$\frac{\left(9+28i\right)\left(92-8i\right)}{92^{2}-8^{2}i^{2}}$$

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

$$\frac{\left(9+28i\right)\left(92-8i\right)}{8528}$$

Multiply complex numbers 9+28i and 92-8i like you multiply binomials.

$$\frac{9\times 92+9\times \left(-8i\right)+28i\times 92+28\left(-8\right)i^{2}}{8528}$$

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

$$\frac{9\times 92+9\times \left(-8i\right)+28i\times 92+28\left(-8\right)\left(-1\right)}{8528}$$

Do the multiplications in 9\times 92+9\times \left(-8i\right)+28i\times 92+28\left(-8\right)\left(-1\right).

$$\frac{828-72i+2576i+224}{8528}$$

Combine the real and imaginary parts in 828-72i+2576i+224.

$$\frac{828+224+\left(-72+2576\right)i}{8528}$$

Do the additions in 828+224+\left(-72+2576\right)i.

$$\frac{1052+2504i}{8528}$$

Divide 1052+2504i by 8528 to get \frac{263}{2132}+\frac{313}{1066}i.

$$\frac{263}{2132}+\frac{313}{1066}i$$

Real Part

$$\frac{263}{2132} = 0.12335834896810506$$

Solution Steps

Multiply both numerator and denominator of \frac{9+28i}{92+8i} by the complex conjugate of the denominator, 92-8i.

$$Re(\frac{\left(9+28i\right)\left(92-8i\right)}{\left(92+8i\right)\left(92-8i\right)})$$

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

$$Re(\frac{\left(9+28i\right)\left(92-8i\right)}{92^{2}-8^{2}i^{2}})$$

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

$$Re(\frac{\left(9+28i\right)\left(92-8i\right)}{8528})$$

Multiply complex numbers 9+28i and 92-8i like you multiply binomials.

$$Re(\frac{9\times 92+9\times \left(-8i\right)+28i\times 92+28\left(-8\right)i^{2}}{8528})$$

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

$$Re(\frac{9\times 92+9\times \left(-8i\right)+28i\times 92+28\left(-8\right)\left(-1\right)}{8528})$$

Do the multiplications in 9\times 92+9\times \left(-8i\right)+28i\times 92+28\left(-8\right)\left(-1\right).

$$Re(\frac{828-72i+2576i+224}{8528})$$

Combine the real and imaginary parts in 828-72i+2576i+224.

$$Re(\frac{828+224+\left(-72+2576\right)i}{8528})$$

Do the additions in 828+224+\left(-72+2576\right)i.

$$Re(\frac{1052+2504i}{8528})$$

Divide 1052+2504i by 8528 to get \frac{263}{2132}+\frac{313}{1066}i.

$$Re(\frac{263}{2132}+\frac{313}{1066}i)$$

The real part of \frac{263}{2132}+\frac{313}{1066}i is \frac{263}{2132}.

$$\frac{263}{2132}$$