Solve 4-8i/2-3i

$$\frac { 4 - 8 i } { 2 - 3 i }$$

Evaluate

$$\frac{32}{13}-\frac{4}{13}i\approx 2.461538462-0.307692308i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2+3i.

$$\frac{\left(4-8i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}$$

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

$$\frac{\left(4-8i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(4-8i\right)\left(2+3i\right)}{13}$$

Multiply complex numbers 4-8i and 2+3i like you multiply binomials.

$$\frac{4\times 2+4\times \left(3i\right)-8i\times 2-8\times 3i^{2}}{13}$$

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

$$\frac{4\times 2+4\times \left(3i\right)-8i\times 2-8\times 3\left(-1\right)}{13}$$

Do the multiplications in 4\times 2+4\times \left(3i\right)-8i\times 2-8\times 3\left(-1\right).

$$\frac{8+12i-16i+24}{13}$$

Combine the real and imaginary parts in 8+12i-16i+24.

$$\frac{8+24+\left(12-16\right)i}{13}$$

Do the additions in 8+24+\left(12-16\right)i.

$$\frac{32-4i}{13}$$

Divide 32-4i by 13 to get \frac{32}{13}-\frac{4}{13}i.

$$\frac{32}{13}-\frac{4}{13}i$$

Real Part

$$\frac{32}{13} = 2\frac{6}{13} = 2.4615384615384617$$

Solution Steps

Multiply both numerator and denominator of \frac{4-8i}{2-3i} by the complex conjugate of the denominator, 2+3i.

$$Re(\frac{\left(4-8i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)})$$

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

$$Re(\frac{\left(4-8i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(4-8i\right)\left(2+3i\right)}{13})$$

Multiply complex numbers 4-8i and 2+3i like you multiply binomials.

$$Re(\frac{4\times 2+4\times \left(3i\right)-8i\times 2-8\times 3i^{2}}{13})$$

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

$$Re(\frac{4\times 2+4\times \left(3i\right)-8i\times 2-8\times 3\left(-1\right)}{13})$$

Do the multiplications in 4\times 2+4\times \left(3i\right)-8i\times 2-8\times 3\left(-1\right).

$$Re(\frac{8+12i-16i+24}{13})$$

Combine the real and imaginary parts in 8+12i-16i+24.

$$Re(\frac{8+24+\left(12-16\right)i}{13})$$

Do the additions in 8+24+\left(12-16\right)i.

$$Re(\frac{32-4i}{13})$$

Divide 32-4i by 13 to get \frac{32}{13}-\frac{4}{13}i.

$$Re(\frac{32}{13}-\frac{4}{13}i)$$

The real part of \frac{32}{13}-\frac{4}{13}i is \frac{32}{13}.

$$\frac{32}{13}$$