Solve 3-5i/3+5i

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

Evaluate

$$-\frac{8}{17}-\frac{15}{17}i\approx -0.470588235-0.882352941i$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$\frac{3\times 3+3\times \left(-5i\right)-5i\times 3-5\left(-5\right)\left(-1\right)}{34}$$

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

$$\frac{9-15i-15i-25}{34}$$

Combine the real and imaginary parts in 9-15i-15i-25.

$$\frac{9-25+\left(-15-15\right)i}{34}$$

Do the additions in 9-25+\left(-15-15\right)i.

$$\frac{-16-30i}{34}$$

Divide -16-30i by 34 to get -\frac{8}{17}-\frac{15}{17}i.

$$-\frac{8}{17}-\frac{15}{17}i$$

Real Part

$$-\frac{8}{17} = -0.47058823529411764$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$Re(\frac{3\times 3+3\times \left(-5i\right)-5i\times 3-5\left(-5\right)\left(-1\right)}{34})$$

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

$$Re(\frac{9-15i-15i-25}{34})$$

Combine the real and imaginary parts in 9-15i-15i-25.

$$Re(\frac{9-25+\left(-15-15\right)i}{34})$$

Do the additions in 9-25+\left(-15-15\right)i.

$$Re(\frac{-16-30i}{34})$$

Divide -16-30i by 34 to get -\frac{8}{17}-\frac{15}{17}i.

$$Re(-\frac{8}{17}-\frac{15}{17}i)$$

The real part of -\frac{8}{17}-\frac{15}{17}i is -\frac{8}{17}.

$$-\frac{8}{17}$$