Solve (3+2i)/4-2i

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

Evaluate

$$\frac{2}{5}+\frac{7}{10}i=0.4+0.7i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{12+6i+8i-4}{20}$$

Combine the real and imaginary parts in 12+6i+8i-4.

$$\frac{12-4+\left(6+8\right)i}{20}$$

Do the additions in 12-4+\left(6+8\right)i.

$$\frac{8+14i}{20}$$

Divide 8+14i by 20 to get \frac{2}{5}+\frac{7}{10}i.

$$\frac{2}{5}+\frac{7}{10}i$$

Real Part

$$\frac{2}{5} = 0.4$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{12+6i+8i-4}{20})$$

Combine the real and imaginary parts in 12+6i+8i-4.

$$Re(\frac{12-4+\left(6+8\right)i}{20})$$

Do the additions in 12-4+\left(6+8\right)i.

$$Re(\frac{8+14i}{20})$$

Divide 8+14i by 20 to get \frac{2}{5}+\frac{7}{10}i.

$$Re(\frac{2}{5}+\frac{7}{10}i)$$

The real part of \frac{2}{5}+\frac{7}{10}i is \frac{2}{5}.

$$\frac{2}{5}$$