Solve 8-4i/3+8i

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

Evaluate

$$-\frac{8}{73}-\frac{76}{73}i\approx -0.109589041-1.04109589i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-8i.

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

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

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

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

$$\frac{\left(8-4i\right)\left(3-8i\right)}{73}$$

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

$$\frac{8\times 3+8\times \left(-8i\right)-4i\times 3-4\left(-8\right)i^{2}}{73}$$

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

$$\frac{8\times 3+8\times \left(-8i\right)-4i\times 3-4\left(-8\right)\left(-1\right)}{73}$$

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

$$\frac{24-64i-12i-32}{73}$$

Combine the real and imaginary parts in 24-64i-12i-32.

$$\frac{24-32+\left(-64-12\right)i}{73}$$

Do the additions in 24-32+\left(-64-12\right)i.

$$\frac{-8-76i}{73}$$

Divide -8-76i by 73 to get -\frac{8}{73}-\frac{76}{73}i.

$$-\frac{8}{73}-\frac{76}{73}i$$

Real Part

$$-\frac{8}{73} = -0.1095890410958904$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(8-4i\right)\left(3-8i\right)}{73})$$

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

$$Re(\frac{8\times 3+8\times \left(-8i\right)-4i\times 3-4\left(-8\right)i^{2}}{73})$$

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

$$Re(\frac{8\times 3+8\times \left(-8i\right)-4i\times 3-4\left(-8\right)\left(-1\right)}{73})$$

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

$$Re(\frac{24-64i-12i-32}{73})$$

Combine the real and imaginary parts in 24-64i-12i-32.

$$Re(\frac{24-32+\left(-64-12\right)i}{73})$$

Do the additions in 24-32+\left(-64-12\right)i.

$$Re(\frac{-8-76i}{73})$$

Divide -8-76i by 73 to get -\frac{8}{73}-\frac{76}{73}i.

$$Re(-\frac{8}{73}-\frac{76}{73}i)$$

The real part of -\frac{8}{73}-\frac{76}{73}i is -\frac{8}{73}.

$$-\frac{8}{73}$$