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

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

Evaluate

$$6-2i$$

Solution Steps

Multiply 10-10i times 10.

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

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

$$\frac{100-100i}{20-10i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 20+10i.

$$\frac{\left(100-100i\right)\left(20+10i\right)}{\left(20-10i\right)\left(20+10i\right)}$$

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

$$\frac{\left(100-100i\right)\left(20+10i\right)}{20^{2}-10^{2}i^{2}}$$

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

$$\frac{\left(100-100i\right)\left(20+10i\right)}{500}$$

Multiply complex numbers 100-100i and 20+10i like you multiply binomials.

$$\frac{100\times 20+100\times \left(10i\right)-100i\times 20-100\times 10i^{2}}{500}$$

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

$$\frac{100\times 20+100\times \left(10i\right)-100i\times 20-100\times 10\left(-1\right)}{500}$$

Do the multiplications in 100\times 20+100\times \left(10i\right)-100i\times 20-100\times 10\left(-1\right).

$$\frac{2000+1000i-2000i+1000}{500}$$

Combine the real and imaginary parts in 2000+1000i-2000i+1000.

$$\frac{2000+1000+\left(1000-2000\right)i}{500}$$

Do the additions in 2000+1000+\left(1000-2000\right)i.

$$\frac{3000-1000i}{500}$$

Divide 3000-1000i by 500 to get 6-2i.

$$6-2i$$

Real Part

$$6$$

Solution Steps

Multiply 10-10i times 10.

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

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

$$Re(\frac{100-100i}{20-10i})$$

Multiply both numerator and denominator of \frac{100-100i}{20-10i} by the complex conjugate of the denominator, 20+10i.

$$Re(\frac{\left(100-100i\right)\left(20+10i\right)}{\left(20-10i\right)\left(20+10i\right)})$$

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

$$Re(\frac{\left(100-100i\right)\left(20+10i\right)}{20^{2}-10^{2}i^{2}})$$

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

$$Re(\frac{\left(100-100i\right)\left(20+10i\right)}{500})$$

Multiply complex numbers 100-100i and 20+10i like you multiply binomials.

$$Re(\frac{100\times 20+100\times \left(10i\right)-100i\times 20-100\times 10i^{2}}{500})$$

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

$$Re(\frac{100\times 20+100\times \left(10i\right)-100i\times 20-100\times 10\left(-1\right)}{500})$$

Do the multiplications in 100\times 20+100\times \left(10i\right)-100i\times 20-100\times 10\left(-1\right).

$$Re(\frac{2000+1000i-2000i+1000}{500})$$

Combine the real and imaginary parts in 2000+1000i-2000i+1000.

$$Re(\frac{2000+1000+\left(1000-2000\right)i}{500})$$

Do the additions in 2000+1000+\left(1000-2000\right)i.

$$Re(\frac{3000-1000i}{500})$$

Divide 3000-1000i by 500 to get 6-2i.

$$Re(6-2i)$$

The real part of 6-2i is 6.

$$6$$