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.

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

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

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

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

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

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

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

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

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