Solve frac{90-30texttt{i}}{70+60texttt{i}}

$$\frac{ 90-30 \texttt{i} }{ 70+60 \texttt{i} }$$

Evaluate

$$\frac{9}{17}-\frac{15}{17}i\approx 0.529411765-0.882352941i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 70-60i.

$$\frac{\left(90-30i\right)\left(70-60i\right)}{\left(70+60i\right)\left(70-60i\right)}$$

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

$$\frac{\left(90-30i\right)\left(70-60i\right)}{70^{2}-60^{2}i^{2}}$$

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

$$\frac{\left(90-30i\right)\left(70-60i\right)}{8500}$$

Multiply complex numbers 90-30i and 70-60i like you multiply binomials.

$$\frac{90\times 70+90\times \left(-60i\right)-30i\times 70-30\left(-60\right)i^{2}}{8500}$$

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

$$\frac{90\times 70+90\times \left(-60i\right)-30i\times 70-30\left(-60\right)\left(-1\right)}{8500}$$

Do the multiplications in 90\times 70+90\times \left(-60i\right)-30i\times 70-30\left(-60\right)\left(-1\right).

$$\frac{6300-5400i-2100i-1800}{8500}$$

Combine the real and imaginary parts in 6300-5400i-2100i-1800.

$$\frac{6300-1800+\left(-5400-2100\right)i}{8500}$$

Do the additions in 6300-1800+\left(-5400-2100\right)i.

$$\frac{4500-7500i}{8500}$$

Divide 4500-7500i by 8500 to get \frac{9}{17}-\frac{15}{17}i.

$$\frac{9}{17}-\frac{15}{17}i$$

Real Part

$$\frac{9}{17} = 0.5294117647058824$$

Solution Steps

Multiply both numerator and denominator of \frac{90-30i}{70+60i} by the complex conjugate of the denominator, 70-60i.

$$Re(\frac{\left(90-30i\right)\left(70-60i\right)}{\left(70+60i\right)\left(70-60i\right)})$$

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

$$Re(\frac{\left(90-30i\right)\left(70-60i\right)}{70^{2}-60^{2}i^{2}})$$

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

$$Re(\frac{\left(90-30i\right)\left(70-60i\right)}{8500})$$

Multiply complex numbers 90-30i and 70-60i like you multiply binomials.

$$Re(\frac{90\times 70+90\times \left(-60i\right)-30i\times 70-30\left(-60\right)i^{2}}{8500})$$

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

$$Re(\frac{90\times 70+90\times \left(-60i\right)-30i\times 70-30\left(-60\right)\left(-1\right)}{8500})$$

Do the multiplications in 90\times 70+90\times \left(-60i\right)-30i\times 70-30\left(-60\right)\left(-1\right).

$$Re(\frac{6300-5400i-2100i-1800}{8500})$$

Combine the real and imaginary parts in 6300-5400i-2100i-1800.

$$Re(\frac{6300-1800+\left(-5400-2100\right)i}{8500})$$

Do the additions in 6300-1800+\left(-5400-2100\right)i.

$$Re(\frac{4500-7500i}{8500})$$

Divide 4500-7500i by 8500 to get \frac{9}{17}-\frac{15}{17}i.

$$Re(\frac{9}{17}-\frac{15}{17}i)$$

The real part of \frac{9}{17}-\frac{15}{17}i is \frac{9}{17}.

$$\frac{9}{17}$$