Solve -4+4i/4-6i

$$\frac { - 4 + 4 i } { 4 - 6 i }$$

Evaluate

$$-\frac{10}{13}-\frac{2}{13}i\approx -0.769230769-0.153846154i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+6i.

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

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

$$\frac{\left(-4+4i\right)\left(4+6i\right)}{4^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(-4+4i\right)\left(4+6i\right)}{52}$$

Multiply complex numbers -4+4i and 4+6i like you multiply binomials.

$$\frac{-4\times 4-4\times \left(6i\right)+4i\times 4+4\times 6i^{2}}{52}$$

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

$$\frac{-4\times 4-4\times \left(6i\right)+4i\times 4+4\times 6\left(-1\right)}{52}$$

Do the multiplications in -4\times 4-4\times \left(6i\right)+4i\times 4+4\times 6\left(-1\right).

$$\frac{-16-24i+16i-24}{52}$$

Combine the real and imaginary parts in -16-24i+16i-24.

$$\frac{-16-24+\left(-24+16\right)i}{52}$$

Do the additions in -16-24+\left(-24+16\right)i.

$$\frac{-40-8i}{52}$$

Divide -40-8i by 52 to get -\frac{10}{13}-\frac{2}{13}i.

$$-\frac{10}{13}-\frac{2}{13}i$$

Real Part

$$-\frac{10}{13} = -0.7692307692307693$$

Solution Steps

Multiply both numerator and denominator of \frac{-4+4i}{4-6i} by the complex conjugate of the denominator, 4+6i.

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

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

$$Re(\frac{\left(-4+4i\right)\left(4+6i\right)}{4^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(-4+4i\right)\left(4+6i\right)}{52})$$

Multiply complex numbers -4+4i and 4+6i like you multiply binomials.

$$Re(\frac{-4\times 4-4\times \left(6i\right)+4i\times 4+4\times 6i^{2}}{52})$$

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

$$Re(\frac{-4\times 4-4\times \left(6i\right)+4i\times 4+4\times 6\left(-1\right)}{52})$$

Do the multiplications in -4\times 4-4\times \left(6i\right)+4i\times 4+4\times 6\left(-1\right).

$$Re(\frac{-16-24i+16i-24}{52})$$

Combine the real and imaginary parts in -16-24i+16i-24.

$$Re(\frac{-16-24+\left(-24+16\right)i}{52})$$

Do the additions in -16-24+\left(-24+16\right)i.

$$Re(\frac{-40-8i}{52})$$

Divide -40-8i by 52 to get -\frac{10}{13}-\frac{2}{13}i.

$$Re(-\frac{10}{13}-\frac{2}{13}i)$$

The real part of -\frac{10}{13}-\frac{2}{13}i is -\frac{10}{13}.

$$-\frac{10}{13}$$