Solve 11-7i/1-2i

$$\frac { 11 - 7 i } { 1 - 2 i }$$

Evaluate

$$5+3i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+2i.

$$\frac{\left(11-7i\right)\left(1+2i\right)}{\left(1-2i\right)\left(1+2i\right)}$$

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

$$\frac{\left(11-7i\right)\left(1+2i\right)}{1^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(11-7i\right)\left(1+2i\right)}{5}$$

Multiply complex numbers 11-7i and 1+2i like you multiply binomials.

$$\frac{11\times 1+11\times \left(2i\right)-7i-7\times 2i^{2}}{5}$$

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

$$\frac{11\times 1+11\times \left(2i\right)-7i-7\times 2\left(-1\right)}{5}$$

Do the multiplications in 11\times 1+11\times \left(2i\right)-7i-7\times 2\left(-1\right).

$$\frac{11+22i-7i+14}{5}$$

Combine the real and imaginary parts in 11+22i-7i+14.

$$\frac{11+14+\left(22-7\right)i}{5}$$

Do the additions in 11+14+\left(22-7\right)i.

$$\frac{25+15i}{5}$$

Divide 25+15i by 5 to get 5+3i.

$$5+3i$$

Real Part

$$5$$

Solution Steps

Multiply both numerator and denominator of \frac{11-7i}{1-2i} by the complex conjugate of the denominator, 1+2i.

$$Re(\frac{\left(11-7i\right)\left(1+2i\right)}{\left(1-2i\right)\left(1+2i\right)})$$

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

$$Re(\frac{\left(11-7i\right)\left(1+2i\right)}{1^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(11-7i\right)\left(1+2i\right)}{5})$$

Multiply complex numbers 11-7i and 1+2i like you multiply binomials.

$$Re(\frac{11\times 1+11\times \left(2i\right)-7i-7\times 2i^{2}}{5})$$

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

$$Re(\frac{11\times 1+11\times \left(2i\right)-7i-7\times 2\left(-1\right)}{5})$$

Do the multiplications in 11\times 1+11\times \left(2i\right)-7i-7\times 2\left(-1\right).

$$Re(\frac{11+22i-7i+14}{5})$$

Combine the real and imaginary parts in 11+22i-7i+14.

$$Re(\frac{11+14+\left(22-7\right)i}{5})$$

Do the additions in 11+14+\left(22-7\right)i.

$$Re(\frac{25+15i}{5})$$

Divide 25+15i by 5 to get 5+3i.

$$Re(5+3i)$$

The real part of 5+3i is 5.

$$5$$