Solve frac{(60-60texttt{i})*20texttt{i}}{60-40texttt{i}}

$$\frac{ (60-60 \texttt{i} ) \times 20 \texttt{i} }{ 60-40 \texttt{i} }$$

Evaluate

$$\frac{60}{13}+\frac{300}{13}i\approx 4.615384615+23.076923077i$$

Solution Steps

Multiply 60-60i times 20i.

$$\frac{60\times \left(20i\right)-60\times 20i^{2}}{60-40i}$$

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

$$\frac{60\times \left(20i\right)-60\times 20\left(-1\right)}{60-40i}$$

Do the multiplications in 60\times \left(20i\right)-60\times 20\left(-1\right). Reorder the terms.

$$\frac{1200+1200i}{60-40i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 60+40i.

$$\frac{\left(1200+1200i\right)\left(60+40i\right)}{\left(60-40i\right)\left(60+40i\right)}$$

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

$$\frac{\left(1200+1200i\right)\left(60+40i\right)}{60^{2}-40^{2}i^{2}}$$

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

$$\frac{\left(1200+1200i\right)\left(60+40i\right)}{5200}$$

Multiply complex numbers 1200+1200i and 60+40i like you multiply binomials.

$$\frac{1200\times 60+1200\times \left(40i\right)+1200i\times 60+1200\times 40i^{2}}{5200}$$

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

$$\frac{1200\times 60+1200\times \left(40i\right)+1200i\times 60+1200\times 40\left(-1\right)}{5200}$$

Do the multiplications in 1200\times 60+1200\times \left(40i\right)+1200i\times 60+1200\times 40\left(-1\right).

$$\frac{72000+48000i+72000i-48000}{5200}$$

Combine the real and imaginary parts in 72000+48000i+72000i-48000.

$$\frac{72000-48000+\left(48000+72000\right)i}{5200}$$

Do the additions in 72000-48000+\left(48000+72000\right)i.

$$\frac{24000+120000i}{5200}$$

Divide 24000+120000i by 5200 to get \frac{60}{13}+\frac{300}{13}i.

$$\frac{60}{13}+\frac{300}{13}i$$

Real Part

$$\frac{60}{13} = 4\frac{8}{13} = 4.615384615384615$$

Solution Steps

Multiply 60-60i times 20i.

$$Re(\frac{60\times \left(20i\right)-60\times 20i^{2}}{60-40i})$$

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

$$Re(\frac{60\times \left(20i\right)-60\times 20\left(-1\right)}{60-40i})$$

Do the multiplications in 60\times \left(20i\right)-60\times 20\left(-1\right). Reorder the terms.

$$Re(\frac{1200+1200i}{60-40i})$$

Multiply both numerator and denominator of \frac{1200+1200i}{60-40i} by the complex conjugate of the denominator, 60+40i.

$$Re(\frac{\left(1200+1200i\right)\left(60+40i\right)}{\left(60-40i\right)\left(60+40i\right)})$$

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

$$Re(\frac{\left(1200+1200i\right)\left(60+40i\right)}{60^{2}-40^{2}i^{2}})$$

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

$$Re(\frac{\left(1200+1200i\right)\left(60+40i\right)}{5200})$$

Multiply complex numbers 1200+1200i and 60+40i like you multiply binomials.

$$Re(\frac{1200\times 60+1200\times \left(40i\right)+1200i\times 60+1200\times 40i^{2}}{5200})$$

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

$$Re(\frac{1200\times 60+1200\times \left(40i\right)+1200i\times 60+1200\times 40\left(-1\right)}{5200})$$

Do the multiplications in 1200\times 60+1200\times \left(40i\right)+1200i\times 60+1200\times 40\left(-1\right).

$$Re(\frac{72000+48000i+72000i-48000}{5200})$$

Combine the real and imaginary parts in 72000+48000i+72000i-48000.

$$Re(\frac{72000-48000+\left(48000+72000\right)i}{5200})$$

Do the additions in 72000-48000+\left(48000+72000\right)i.

$$Re(\frac{24000+120000i}{5200})$$

Divide 24000+120000i by 5200 to get \frac{60}{13}+\frac{300}{13}i.

$$Re(\frac{60}{13}+\frac{300}{13}i)$$

The real part of \frac{60}{13}+\frac{300}{13}i is \frac{60}{13}.

$$\frac{60}{13}$$