Solve 5-3i/4+2i=

$$\frac { 5 - 3 i } { 4 + 2 i } =$$

Evaluate

$$\frac{7}{10}-\frac{11}{10}i=0.7-1.1i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4-2i.

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

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

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

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

$$\frac{\left(5-3i\right)\left(4-2i\right)}{20}$$

Multiply complex numbers 5-3i and 4-2i like you multiply binomials.

$$\frac{5\times 4+5\times \left(-2i\right)-3i\times 4-3\left(-2\right)i^{2}}{20}$$

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

$$\frac{5\times 4+5\times \left(-2i\right)-3i\times 4-3\left(-2\right)\left(-1\right)}{20}$$

Do the multiplications in 5\times 4+5\times \left(-2i\right)-3i\times 4-3\left(-2\right)\left(-1\right).

$$\frac{20-10i-12i-6}{20}$$

Combine the real and imaginary parts in 20-10i-12i-6.

$$\frac{20-6+\left(-10-12\right)i}{20}$$

Do the additions in 20-6+\left(-10-12\right)i.

$$\frac{14-22i}{20}$$

Divide 14-22i by 20 to get \frac{7}{10}-\frac{11}{10}i.

$$\frac{7}{10}-\frac{11}{10}i$$

Real Part

$$\frac{7}{10} = 0.7$$

Solution Steps

Multiply both numerator and denominator of \frac{5-3i}{4+2i} by the complex conjugate of the denominator, 4-2i.

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

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

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

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

$$Re(\frac{\left(5-3i\right)\left(4-2i\right)}{20})$$

Multiply complex numbers 5-3i and 4-2i like you multiply binomials.

$$Re(\frac{5\times 4+5\times \left(-2i\right)-3i\times 4-3\left(-2\right)i^{2}}{20})$$

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

$$Re(\frac{5\times 4+5\times \left(-2i\right)-3i\times 4-3\left(-2\right)\left(-1\right)}{20})$$

Do the multiplications in 5\times 4+5\times \left(-2i\right)-3i\times 4-3\left(-2\right)\left(-1\right).

$$Re(\frac{20-10i-12i-6}{20})$$

Combine the real and imaginary parts in 20-10i-12i-6.

$$Re(\frac{20-6+\left(-10-12\right)i}{20})$$

Do the additions in 20-6+\left(-10-12\right)i.

$$Re(\frac{14-22i}{20})$$

Divide 14-22i by 20 to get \frac{7}{10}-\frac{11}{10}i.

$$Re(\frac{7}{10}-\frac{11}{10}i)$$

The real part of \frac{7}{10}-\frac{11}{10}i is \frac{7}{10}.

$$\frac{7}{10}$$