Solve 8+4i/9-3i

$$\frac { 8 + 4 i } { 9 - 3 i }$$

Evaluate

$$\frac{2}{3}+\frac{2}{3}i\approx 0.666666667+0.666666667i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 9+3i.

$$\frac{\left(8+4i\right)\left(9+3i\right)}{\left(9-3i\right)\left(9+3i\right)}$$

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

$$\frac{\left(8+4i\right)\left(9+3i\right)}{9^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(8+4i\right)\left(9+3i\right)}{90}$$

Multiply complex numbers 8+4i and 9+3i like you multiply binomials.

$$\frac{8\times 9+8\times \left(3i\right)+4i\times 9+4\times 3i^{2}}{90}$$

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

$$\frac{8\times 9+8\times \left(3i\right)+4i\times 9+4\times 3\left(-1\right)}{90}$$

Do the multiplications in 8\times 9+8\times \left(3i\right)+4i\times 9+4\times 3\left(-1\right).

$$\frac{72+24i+36i-12}{90}$$

Combine the real and imaginary parts in 72+24i+36i-12.

$$\frac{72-12+\left(24+36\right)i}{90}$$

Do the additions in 72-12+\left(24+36\right)i.

$$\frac{60+60i}{90}$$

Divide 60+60i by 90 to get \frac{2}{3}+\frac{2}{3}i.

$$\frac{2}{3}+\frac{2}{3}i$$

Real Part

$$\frac{2}{3} = 0.6666666666666666$$

Solution Steps

Multiply both numerator and denominator of \frac{8+4i}{9-3i} by the complex conjugate of the denominator, 9+3i.

$$Re(\frac{\left(8+4i\right)\left(9+3i\right)}{\left(9-3i\right)\left(9+3i\right)})$$

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

$$Re(\frac{\left(8+4i\right)\left(9+3i\right)}{9^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(8+4i\right)\left(9+3i\right)}{90})$$

Multiply complex numbers 8+4i and 9+3i like you multiply binomials.

$$Re(\frac{8\times 9+8\times \left(3i\right)+4i\times 9+4\times 3i^{2}}{90})$$

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

$$Re(\frac{8\times 9+8\times \left(3i\right)+4i\times 9+4\times 3\left(-1\right)}{90})$$

Do the multiplications in 8\times 9+8\times \left(3i\right)+4i\times 9+4\times 3\left(-1\right).

$$Re(\frac{72+24i+36i-12}{90})$$

Combine the real and imaginary parts in 72+24i+36i-12.

$$Re(\frac{72-12+\left(24+36\right)i}{90})$$

Do the additions in 72-12+\left(24+36\right)i.

$$Re(\frac{60+60i}{90})$$

Divide 60+60i by 90 to get \frac{2}{3}+\frac{2}{3}i.

$$Re(\frac{2}{3}+\frac{2}{3}i)$$

The real part of \frac{2}{3}+\frac{2}{3}i is \frac{2}{3}.

$$\frac{2}{3}$$