Solve 5-8i/3+6i

$$\frac { 5 - 8 i } { 3 + 6 i }$$

Evaluate

$$-\frac{11}{15}-\frac{6}{5}i\approx -0.733333333-1.2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-6i.

$$\frac{\left(5-8i\right)\left(3-6i\right)}{\left(3+6i\right)\left(3-6i\right)}$$

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

$$\frac{\left(5-8i\right)\left(3-6i\right)}{3^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(5-8i\right)\left(3-6i\right)}{45}$$

Multiply complex numbers 5-8i and 3-6i like you multiply binomials.

$$\frac{5\times 3+5\times \left(-6i\right)-8i\times 3-8\left(-6\right)i^{2}}{45}$$

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

$$\frac{5\times 3+5\times \left(-6i\right)-8i\times 3-8\left(-6\right)\left(-1\right)}{45}$$

Do the multiplications in 5\times 3+5\times \left(-6i\right)-8i\times 3-8\left(-6\right)\left(-1\right).

$$\frac{15-30i-24i-48}{45}$$

Combine the real and imaginary parts in 15-30i-24i-48.

$$\frac{15-48+\left(-30-24\right)i}{45}$$

Do the additions in 15-48+\left(-30-24\right)i.

$$\frac{-33-54i}{45}$$

Divide -33-54i by 45 to get -\frac{11}{15}-\frac{6}{5}i.

$$-\frac{11}{15}-\frac{6}{5}i$$

Real Part

$$-\frac{11}{15} = -0.7333333333333333$$

Solution Steps

Multiply both numerator and denominator of \frac{5-8i}{3+6i} by the complex conjugate of the denominator, 3-6i.

$$Re(\frac{\left(5-8i\right)\left(3-6i\right)}{\left(3+6i\right)\left(3-6i\right)})$$

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

$$Re(\frac{\left(5-8i\right)\left(3-6i\right)}{3^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(5-8i\right)\left(3-6i\right)}{45})$$

Multiply complex numbers 5-8i and 3-6i like you multiply binomials.

$$Re(\frac{5\times 3+5\times \left(-6i\right)-8i\times 3-8\left(-6\right)i^{2}}{45})$$

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

$$Re(\frac{5\times 3+5\times \left(-6i\right)-8i\times 3-8\left(-6\right)\left(-1\right)}{45})$$

Do the multiplications in 5\times 3+5\times \left(-6i\right)-8i\times 3-8\left(-6\right)\left(-1\right).

$$Re(\frac{15-30i-24i-48}{45})$$

Combine the real and imaginary parts in 15-30i-24i-48.

$$Re(\frac{15-48+\left(-30-24\right)i}{45})$$

Do the additions in 15-48+\left(-30-24\right)i.

$$Re(\frac{-33-54i}{45})$$

Divide -33-54i by 45 to get -\frac{11}{15}-\frac{6}{5}i.

$$Re(-\frac{11}{15}-\frac{6}{5}i)$$

The real part of -\frac{11}{15}-\frac{6}{5}i is -\frac{11}{15}.

$$-\frac{11}{15}$$