Solve 2/5-3i

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

Evaluate

$$\frac{5}{17}+\frac{3}{17}i\approx 0.294117647+0.176470588i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5+3i.

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

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

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

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

$$\frac{2\left(5+3i\right)}{34}$$

Multiply 2 times 5+3i.

$$\frac{2\times 5+2\times \left(3i\right)}{34}$$

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

$$\frac{10+6i}{34}$$

Divide 10+6i by 34 to get \frac{5}{17}+\frac{3}{17}i.

$$\frac{5}{17}+\frac{3}{17}i$$

Real Part

$$\frac{5}{17} = 0.29411764705882354$$

Solution Steps

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

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

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

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

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

$$Re(\frac{2\left(5+3i\right)}{34})$$

Multiply 2 times 5+3i.

$$Re(\frac{2\times 5+2\times \left(3i\right)}{34})$$

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

$$Re(\frac{10+6i}{34})$$

Divide 10+6i by 34 to get \frac{5}{17}+\frac{3}{17}i.

$$Re(\frac{5}{17}+\frac{3}{17}i)$$

The real part of \frac{5}{17}+\frac{3}{17}i is \frac{5}{17}.

$$\frac{5}{17}$$