Solve frac{(10-10texttt{i})*10}{20-11texttt{i}}

$$\frac{ (10-10 \texttt{i} ) \times 10 }{ 20-11 \texttt{i} }$$

Evaluate

$$\frac{3100}{521}-\frac{900}{521}i\approx 5.950095969-1.727447217i$$

Solution Steps

Multiply 10-10i times 10.

$$\frac{10\times 10-10i\times 10}{20-11i}$$

Do the multiplications in 10\times 10-10i\times 10.

$$\frac{100-100i}{20-11i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 20+11i.

$$\frac{\left(100-100i\right)\left(20+11i\right)}{\left(20-11i\right)\left(20+11i\right)}$$

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

$$\frac{\left(100-100i\right)\left(20+11i\right)}{20^{2}-11^{2}i^{2}}$$

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

$$\frac{\left(100-100i\right)\left(20+11i\right)}{521}$$

Multiply complex numbers 100-100i and 20+11i like you multiply binomials.

$$\frac{100\times 20+100\times \left(11i\right)-100i\times 20-100\times 11i^{2}}{521}$$

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

$$\frac{100\times 20+100\times \left(11i\right)-100i\times 20-100\times 11\left(-1\right)}{521}$$

Do the multiplications in 100\times 20+100\times \left(11i\right)-100i\times 20-100\times 11\left(-1\right).

$$\frac{2000+1100i-2000i+1100}{521}$$

Combine the real and imaginary parts in 2000+1100i-2000i+1100.

$$\frac{2000+1100+\left(1100-2000\right)i}{521}$$

Do the additions in 2000+1100+\left(1100-2000\right)i.

$$\frac{3100-900i}{521}$$

Divide 3100-900i by 521 to get \frac{3100}{521}-\frac{900}{521}i.

$$\frac{3100}{521}-\frac{900}{521}i$$

Real Part

$$\frac{3100}{521} = 5\frac{495}{521} = 5.950095969289827$$

Solution Steps

Multiply 10-10i times 10.

$$Re(\frac{10\times 10-10i\times 10}{20-11i})$$

Do the multiplications in 10\times 10-10i\times 10.

$$Re(\frac{100-100i}{20-11i})$$

Multiply both numerator and denominator of \frac{100-100i}{20-11i} by the complex conjugate of the denominator, 20+11i.

$$Re(\frac{\left(100-100i\right)\left(20+11i\right)}{\left(20-11i\right)\left(20+11i\right)})$$

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

$$Re(\frac{\left(100-100i\right)\left(20+11i\right)}{20^{2}-11^{2}i^{2}})$$

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

$$Re(\frac{\left(100-100i\right)\left(20+11i\right)}{521})$$

Multiply complex numbers 100-100i and 20+11i like you multiply binomials.

$$Re(\frac{100\times 20+100\times \left(11i\right)-100i\times 20-100\times 11i^{2}}{521})$$

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

$$Re(\frac{100\times 20+100\times \left(11i\right)-100i\times 20-100\times 11\left(-1\right)}{521})$$

Do the multiplications in 100\times 20+100\times \left(11i\right)-100i\times 20-100\times 11\left(-1\right).

$$Re(\frac{2000+1100i-2000i+1100}{521})$$

Combine the real and imaginary parts in 2000+1100i-2000i+1100.

$$Re(\frac{2000+1100+\left(1100-2000\right)i}{521})$$

Do the additions in 2000+1100+\left(1100-2000\right)i.

$$Re(\frac{3100-900i}{521})$$

Divide 3100-900i by 521 to get \frac{3100}{521}-\frac{900}{521}i.

$$Re(\frac{3100}{521}-\frac{900}{521}i)$$

The real part of \frac{3100}{521}-\frac{900}{521}i is \frac{3100}{521}.

$$\frac{3100}{521}$$