Solve 8+6i/55+10i

$$\frac { 8 + 6 i } { 55 + 10 i }$$

Evaluate

$$\frac{4}{25}+\frac{2}{25}i=0.16+0.08i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 55-10i.

$$\frac{\left(8+6i\right)\left(55-10i\right)}{\left(55+10i\right)\left(55-10i\right)}$$

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

$$\frac{\left(8+6i\right)\left(55-10i\right)}{55^{2}-10^{2}i^{2}}$$

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

$$\frac{\left(8+6i\right)\left(55-10i\right)}{3125}$$

Multiply complex numbers 8+6i and 55-10i like you multiply binomials.

$$\frac{8\times 55+8\times \left(-10i\right)+6i\times 55+6\left(-10\right)i^{2}}{3125}$$

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

$$\frac{8\times 55+8\times \left(-10i\right)+6i\times 55+6\left(-10\right)\left(-1\right)}{3125}$$

Do the multiplications in 8\times 55+8\times \left(-10i\right)+6i\times 55+6\left(-10\right)\left(-1\right).

$$\frac{440-80i+330i+60}{3125}$$

Combine the real and imaginary parts in 440-80i+330i+60.

$$\frac{440+60+\left(-80+330\right)i}{3125}$$

Do the additions in 440+60+\left(-80+330\right)i.

$$\frac{500+250i}{3125}$$

Divide 500+250i by 3125 to get \frac{4}{25}+\frac{2}{25}i.

$$\frac{4}{25}+\frac{2}{25}i$$

Real Part

$$\frac{4}{25} = 0.16$$

Solution Steps

Multiply both numerator and denominator of \frac{8+6i}{55+10i} by the complex conjugate of the denominator, 55-10i.

$$Re(\frac{\left(8+6i\right)\left(55-10i\right)}{\left(55+10i\right)\left(55-10i\right)})$$

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

$$Re(\frac{\left(8+6i\right)\left(55-10i\right)}{55^{2}-10^{2}i^{2}})$$

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

$$Re(\frac{\left(8+6i\right)\left(55-10i\right)}{3125})$$

Multiply complex numbers 8+6i and 55-10i like you multiply binomials.

$$Re(\frac{8\times 55+8\times \left(-10i\right)+6i\times 55+6\left(-10\right)i^{2}}{3125})$$

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

$$Re(\frac{8\times 55+8\times \left(-10i\right)+6i\times 55+6\left(-10\right)\left(-1\right)}{3125})$$

Do the multiplications in 8\times 55+8\times \left(-10i\right)+6i\times 55+6\left(-10\right)\left(-1\right).

$$Re(\frac{440-80i+330i+60}{3125})$$

Combine the real and imaginary parts in 440-80i+330i+60.

$$Re(\frac{440+60+\left(-80+330\right)i}{3125})$$

Do the additions in 440+60+\left(-80+330\right)i.

$$Re(\frac{500+250i}{3125})$$

Divide 500+250i by 3125 to get \frac{4}{25}+\frac{2}{25}i.

$$Re(\frac{4}{25}+\frac{2}{25}i)$$

The real part of \frac{4}{25}+\frac{2}{25}i is \frac{4}{25}.

$$\frac{4}{25}$$