Solve 5-5i/4-3i

$$\frac{ 5-5i }{ 4-3i }$$

Evaluate

$$\frac{7}{5}-\frac{1}{5}i=1.4-0.2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+3i.

$$\frac{\left(5-5i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)}$$

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

$$\frac{\left(5-5i\right)\left(4+3i\right)}{4^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(5-5i\right)\left(4+3i\right)}{25}$$

Multiply complex numbers 5-5i and 4+3i like you multiply binomials.

$$\frac{5\times 4+5\times \left(3i\right)-5i\times 4-5\times 3i^{2}}{25}$$

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

$$\frac{5\times 4+5\times \left(3i\right)-5i\times 4-5\times 3\left(-1\right)}{25}$$

Do the multiplications in 5\times 4+5\times \left(3i\right)-5i\times 4-5\times 3\left(-1\right).

$$\frac{20+15i-20i+15}{25}$$

Combine the real and imaginary parts in 20+15i-20i+15.

$$\frac{20+15+\left(15-20\right)i}{25}$$

Do the additions in 20+15+\left(15-20\right)i.

$$\frac{35-5i}{25}$$

Divide 35-5i by 25 to get \frac{7}{5}-\frac{1}{5}i.

$$\frac{7}{5}-\frac{1}{5}i$$

Real Part

$$\frac{7}{5} = 1\frac{2}{5} = 1.4$$

Solution Steps

Multiply both numerator and denominator of \frac{5-5i}{4-3i} by the complex conjugate of the denominator, 4+3i.

$$Re(\frac{\left(5-5i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)})$$

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

$$Re(\frac{\left(5-5i\right)\left(4+3i\right)}{4^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(5-5i\right)\left(4+3i\right)}{25})$$

Multiply complex numbers 5-5i and 4+3i like you multiply binomials.

$$Re(\frac{5\times 4+5\times \left(3i\right)-5i\times 4-5\times 3i^{2}}{25})$$

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

$$Re(\frac{5\times 4+5\times \left(3i\right)-5i\times 4-5\times 3\left(-1\right)}{25})$$

Do the multiplications in 5\times 4+5\times \left(3i\right)-5i\times 4-5\times 3\left(-1\right).

$$Re(\frac{20+15i-20i+15}{25})$$

Combine the real and imaginary parts in 20+15i-20i+15.

$$Re(\frac{20+15+\left(15-20\right)i}{25})$$

Do the additions in 20+15+\left(15-20\right)i.

$$Re(\frac{35-5i}{25})$$

Divide 35-5i by 25 to get \frac{7}{5}-\frac{1}{5}i.

$$Re(\frac{7}{5}-\frac{1}{5}i)$$

The real part of \frac{7}{5}-\frac{1}{5}i is \frac{7}{5}.

$$\frac{7}{5}$$