Solve 8+2i/4-6i

$$\frac { 8 + 2 i } { 4 - 6 i }$$

Evaluate

$$\frac{5}{13}+\frac{14}{13}i\approx 0.384615385+1.076923077i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+6i.

$$\frac{\left(8+2i\right)\left(4+6i\right)}{\left(4-6i\right)\left(4+6i\right)}$$

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

$$\frac{\left(8+2i\right)\left(4+6i\right)}{4^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(8+2i\right)\left(4+6i\right)}{52}$$

Multiply complex numbers 8+2i and 4+6i like you multiply binomials.

$$\frac{8\times 4+8\times \left(6i\right)+2i\times 4+2\times 6i^{2}}{52}$$

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

$$\frac{8\times 4+8\times \left(6i\right)+2i\times 4+2\times 6\left(-1\right)}{52}$$

Do the multiplications in 8\times 4+8\times \left(6i\right)+2i\times 4+2\times 6\left(-1\right).

$$\frac{32+48i+8i-12}{52}$$

Combine the real and imaginary parts in 32+48i+8i-12.

$$\frac{32-12+\left(48+8\right)i}{52}$$

Do the additions in 32-12+\left(48+8\right)i.

$$\frac{20+56i}{52}$$

Divide 20+56i by 52 to get \frac{5}{13}+\frac{14}{13}i.

$$\frac{5}{13}+\frac{14}{13}i$$

Real Part

$$\frac{5}{13} = 0.38461538461538464$$

Solution Steps

Multiply both numerator and denominator of \frac{8+2i}{4-6i} by the complex conjugate of the denominator, 4+6i.

$$Re(\frac{\left(8+2i\right)\left(4+6i\right)}{\left(4-6i\right)\left(4+6i\right)})$$

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

$$Re(\frac{\left(8+2i\right)\left(4+6i\right)}{4^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(8+2i\right)\left(4+6i\right)}{52})$$

Multiply complex numbers 8+2i and 4+6i like you multiply binomials.

$$Re(\frac{8\times 4+8\times \left(6i\right)+2i\times 4+2\times 6i^{2}}{52})$$

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

$$Re(\frac{8\times 4+8\times \left(6i\right)+2i\times 4+2\times 6\left(-1\right)}{52})$$

Do the multiplications in 8\times 4+8\times \left(6i\right)+2i\times 4+2\times 6\left(-1\right).

$$Re(\frac{32+48i+8i-12}{52})$$

Combine the real and imaginary parts in 32+48i+8i-12.

$$Re(\frac{32-12+\left(48+8\right)i}{52})$$

Do the additions in 32-12+\left(48+8\right)i.

$$Re(\frac{20+56i}{52})$$

Divide 20+56i by 52 to get \frac{5}{13}+\frac{14}{13}i.

$$Re(\frac{5}{13}+\frac{14}{13}i)$$

The real part of \frac{5}{13}+\frac{14}{13}i is \frac{5}{13}.

$$\frac{5}{13}$$