Solve 8-4i/4-2i

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

Evaluate

$$2$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+2i.

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

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

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

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

$$\frac{\left(8-4i\right)\left(4+2i\right)}{20}$$

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

$$\frac{8\times 4+8\times \left(2i\right)-4i\times 4-4\times 2i^{2}}{20}$$

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

$$\frac{8\times 4+8\times \left(2i\right)-4i\times 4-4\times 2\left(-1\right)}{20}$$

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

$$\frac{32+16i-16i+8}{20}$$

Combine the real and imaginary parts in 32+16i-16i+8.

$$\frac{32+8+\left(16-16\right)i}{20}$$

Do the additions in 32+8+\left(16-16\right)i.

$$\frac{40}{20}$$

Divide 40 by 20 to get 2.

$$2$$

Real Part

$$2$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(8-4i\right)\left(4+2i\right)}{20})$$

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

$$Re(\frac{8\times 4+8\times \left(2i\right)-4i\times 4-4\times 2i^{2}}{20})$$

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

$$Re(\frac{8\times 4+8\times \left(2i\right)-4i\times 4-4\times 2\left(-1\right)}{20})$$

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

$$Re(\frac{32+16i-16i+8}{20})$$

Combine the real and imaginary parts in 32+16i-16i+8.

$$Re(\frac{32+8+\left(16-16\right)i}{20})$$

Do the additions in 32+8+\left(16-16\right)i.

$$Re(\frac{40}{20})$$

Divide 40 by 20 to get 2.

$$Re(2)$$

The real part of 2 is 2.

$$2$$