Solve 5+2i/2-3i

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

Evaluate

$$\frac{4}{13}+\frac{19}{13}i\approx 0.307692308+1.461538462i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2+3i.

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

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

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

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

$$\frac{\left(5+2i\right)\left(2+3i\right)}{13}$$

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

$$\frac{5\times 2+5\times \left(3i\right)+2i\times 2+2\times 3i^{2}}{13}$$

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

$$\frac{5\times 2+5\times \left(3i\right)+2i\times 2+2\times 3\left(-1\right)}{13}$$

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

$$\frac{10+15i+4i-6}{13}$$

Combine the real and imaginary parts in 10+15i+4i-6.

$$\frac{10-6+\left(15+4\right)i}{13}$$

Do the additions in 10-6+\left(15+4\right)i.

$$\frac{4+19i}{13}$$

Divide 4+19i by 13 to get \frac{4}{13}+\frac{19}{13}i.

$$\frac{4}{13}+\frac{19}{13}i$$

Real Part

$$\frac{4}{13} = 0.3076923076923077$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(5+2i\right)\left(2+3i\right)}{13})$$

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

$$Re(\frac{5\times 2+5\times \left(3i\right)+2i\times 2+2\times 3i^{2}}{13})$$

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

$$Re(\frac{5\times 2+5\times \left(3i\right)+2i\times 2+2\times 3\left(-1\right)}{13})$$

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

$$Re(\frac{10+15i+4i-6}{13})$$

Combine the real and imaginary parts in 10+15i+4i-6.

$$Re(\frac{10-6+\left(15+4\right)i}{13})$$

Do the additions in 10-6+\left(15+4\right)i.

$$Re(\frac{4+19i}{13})$$

Divide 4+19i by 13 to get \frac{4}{13}+\frac{19}{13}i.

$$Re(\frac{4}{13}+\frac{19}{13}i)$$

The real part of \frac{4}{13}+\frac{19}{13}i is \frac{4}{13}.

$$\frac{4}{13}$$