Solve 4+4i/2+i=

$$\frac { 4 + 4 i } { 2 + i } =$$

Evaluate

$$\frac{12}{5}+\frac{4}{5}i=2.4+0.8i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(4+4i\right)\left(2-i\right)}{5}$$

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

$$\frac{4\times 2+4\left(-i\right)+4i\times 2+4\left(-1\right)i^{2}}{5}$$

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

$$\frac{4\times 2+4\left(-i\right)+4i\times 2+4\left(-1\right)\left(-1\right)}{5}$$

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

$$\frac{8-4i+8i+4}{5}$$

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

$$\frac{8+4+\left(-4+8\right)i}{5}$$

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

$$\frac{12+4i}{5}$$

Divide 12+4i by 5 to get \frac{12}{5}+\frac{4}{5}i.

$$\frac{12}{5}+\frac{4}{5}i$$

Real Part

$$\frac{12}{5} = 2\frac{2}{5} = 2.4$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(4+4i\right)\left(2-i\right)}{5})$$

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

$$Re(\frac{4\times 2+4\left(-i\right)+4i\times 2+4\left(-1\right)i^{2}}{5})$$

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

$$Re(\frac{4\times 2+4\left(-i\right)+4i\times 2+4\left(-1\right)\left(-1\right)}{5})$$

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

$$Re(\frac{8-4i+8i+4}{5})$$

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

$$Re(\frac{8+4+\left(-4+8\right)i}{5})$$

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

$$Re(\frac{12+4i}{5})$$

Divide 12+4i by 5 to get \frac{12}{5}+\frac{4}{5}i.

$$Re(\frac{12}{5}+\frac{4}{5}i)$$

The real part of \frac{12}{5}+\frac{4}{5}i is \frac{12}{5}.

$$\frac{12}{5}$$