Solve 4+4i/4+i

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

Evaluate

$$\frac{20}{17}+\frac{12}{17}i\approx 1.176470588+0.705882353i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(4+4i\right)\left(4-i\right)}{17}$$

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

$$\frac{4\times 4+4\left(-i\right)+4i\times 4+4\left(-1\right)i^{2}}{17}$$

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

$$\frac{4\times 4+4\left(-i\right)+4i\times 4+4\left(-1\right)\left(-1\right)}{17}$$

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

$$\frac{16-4i+16i+4}{17}$$

Combine the real and imaginary parts in 16-4i+16i+4.

$$\frac{16+4+\left(-4+16\right)i}{17}$$

Do the additions in 16+4+\left(-4+16\right)i.

$$\frac{20+12i}{17}$$

Divide 20+12i by 17 to get \frac{20}{17}+\frac{12}{17}i.

$$\frac{20}{17}+\frac{12}{17}i$$

Real Part

$$\frac{20}{17} = 1\frac{3}{17} = 1.1764705882352942$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(4+4i\right)\left(4-i\right)}{17})$$

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

$$Re(\frac{4\times 4+4\left(-i\right)+4i\times 4+4\left(-1\right)i^{2}}{17})$$

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

$$Re(\frac{4\times 4+4\left(-i\right)+4i\times 4+4\left(-1\right)\left(-1\right)}{17})$$

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

$$Re(\frac{16-4i+16i+4}{17})$$

Combine the real and imaginary parts in 16-4i+16i+4.

$$Re(\frac{16+4+\left(-4+16\right)i}{17})$$

Do the additions in 16+4+\left(-4+16\right)i.

$$Re(\frac{20+12i}{17})$$

Divide 20+12i by 17 to get \frac{20}{17}+\frac{12}{17}i.

$$Re(\frac{20}{17}+\frac{12}{17}i)$$

The real part of \frac{20}{17}+\frac{12}{17}i is \frac{20}{17}.

$$\frac{20}{17}$$