Solve 5+3i/2-4i=

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

Evaluate

$$-\frac{1}{10}+\frac{13}{10}i=-0.1+1.3i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{-2+26i}{20}$$

Divide -2+26i by 20 to get -\frac{1}{10}+\frac{13}{10}i.

$$-\frac{1}{10}+\frac{13}{10}i$$

Real Part

$$-\frac{1}{10} = -0.1$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{-2+26i}{20})$$

Divide -2+26i by 20 to get -\frac{1}{10}+\frac{13}{10}i.

$$Re(-\frac{1}{10}+\frac{13}{10}i)$$

The real part of -\frac{1}{10}+\frac{13}{10}i is -\frac{1}{10}.

$$-\frac{1}{10}$$