Solve 3-5i/8+2i

$$\frac { 3 - 5 i } { 8 + 2 i }$$

Evaluate

$$\frac{7}{34}-\frac{23}{34}i\approx 0.205882353-0.676470588i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 8-2i.

$$\frac{\left(3-5i\right)\left(8-2i\right)}{\left(8+2i\right)\left(8-2i\right)}$$

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

$$\frac{\left(3-5i\right)\left(8-2i\right)}{8^{2}-2^{2}i^{2}}$$

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

$$\frac{\left(3-5i\right)\left(8-2i\right)}{68}$$

Multiply complex numbers 3-5i and 8-2i like you multiply binomials.

$$\frac{3\times 8+3\times \left(-2i\right)-5i\times 8-5\left(-2\right)i^{2}}{68}$$

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

$$\frac{3\times 8+3\times \left(-2i\right)-5i\times 8-5\left(-2\right)\left(-1\right)}{68}$$

Do the multiplications in 3\times 8+3\times \left(-2i\right)-5i\times 8-5\left(-2\right)\left(-1\right).

$$\frac{24-6i-40i-10}{68}$$

Combine the real and imaginary parts in 24-6i-40i-10.

$$\frac{24-10+\left(-6-40\right)i}{68}$$

Do the additions in 24-10+\left(-6-40\right)i.

$$\frac{14-46i}{68}$$

Divide 14-46i by 68 to get \frac{7}{34}-\frac{23}{34}i.

$$\frac{7}{34}-\frac{23}{34}i$$

Real Part

$$\frac{7}{34} = 0.20588235294117646$$

Solution Steps

Multiply both numerator and denominator of \frac{3-5i}{8+2i} by the complex conjugate of the denominator, 8-2i.

$$Re(\frac{\left(3-5i\right)\left(8-2i\right)}{\left(8+2i\right)\left(8-2i\right)})$$

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

$$Re(\frac{\left(3-5i\right)\left(8-2i\right)}{8^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{\left(3-5i\right)\left(8-2i\right)}{68})$$

Multiply complex numbers 3-5i and 8-2i like you multiply binomials.

$$Re(\frac{3\times 8+3\times \left(-2i\right)-5i\times 8-5\left(-2\right)i^{2}}{68})$$

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

$$Re(\frac{3\times 8+3\times \left(-2i\right)-5i\times 8-5\left(-2\right)\left(-1\right)}{68})$$

Do the multiplications in 3\times 8+3\times \left(-2i\right)-5i\times 8-5\left(-2\right)\left(-1\right).

$$Re(\frac{24-6i-40i-10}{68})$$

Combine the real and imaginary parts in 24-6i-40i-10.

$$Re(\frac{24-10+\left(-6-40\right)i}{68})$$

Do the additions in 24-10+\left(-6-40\right)i.

$$Re(\frac{14-46i}{68})$$

Divide 14-46i by 68 to get \frac{7}{34}-\frac{23}{34}i.

$$Re(\frac{7}{34}-\frac{23}{34}i)$$

The real part of \frac{7}{34}-\frac{23}{34}i is \frac{7}{34}.

$$\frac{7}{34}$$