Solve 8-i/8+4i

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

Evaluate

$$\frac{3}{4}-\frac{1}{2}i=0.75-0.5i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 8-4i.

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

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

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

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

$$\frac{\left(8-i\right)\left(8-4i\right)}{80}$$

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

$$\frac{8\times 8+8\times \left(-4i\right)-i\times 8-\left(-4i^{2}\right)}{80}$$

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

$$\frac{8\times 8+8\times \left(-4i\right)-i\times 8-\left(-4\left(-1\right)\right)}{80}$$

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

$$\frac{64-32i-8i-4}{80}$$

Combine the real and imaginary parts in 64-32i-8i-4.

$$\frac{64-4+\left(-32-8\right)i}{80}$$

Do the additions in 64-4+\left(-32-8\right)i.

$$\frac{60-40i}{80}$$

Divide 60-40i by 80 to get \frac{3}{4}-\frac{1}{2}i.

$$\frac{3}{4}-\frac{1}{2}i$$

Real Part

$$\frac{3}{4} = 0.75$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(8-i\right)\left(8-4i\right)}{80})$$

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

$$Re(\frac{8\times 8+8\times \left(-4i\right)-i\times 8-\left(-4i^{2}\right)}{80})$$

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

$$Re(\frac{8\times 8+8\times \left(-4i\right)-i\times 8-\left(-4\left(-1\right)\right)}{80})$$

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

$$Re(\frac{64-32i-8i-4}{80})$$

Combine the real and imaginary parts in 64-32i-8i-4.

$$Re(\frac{64-4+\left(-32-8\right)i}{80})$$

Do the additions in 64-4+\left(-32-8\right)i.

$$Re(\frac{60-40i}{80})$$

Divide 60-40i by 80 to get \frac{3}{4}-\frac{1}{2}i.

$$Re(\frac{3}{4}-\frac{1}{2}i)$$

The real part of \frac{3}{4}-\frac{1}{2}i is \frac{3}{4}.

$$\frac{3}{4}$$