Solve 9+6i/3+8i

$$\frac { 9 + 6 i } { 3 + 8 i }$$

Evaluate

$$\frac{75}{73}-\frac{54}{73}i\approx 1.02739726-0.739726027i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-8i.

$$\frac{\left(9+6i\right)\left(3-8i\right)}{\left(3+8i\right)\left(3-8i\right)}$$

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

$$\frac{\left(9+6i\right)\left(3-8i\right)}{3^{2}-8^{2}i^{2}}$$

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

$$\frac{\left(9+6i\right)\left(3-8i\right)}{73}$$

Multiply complex numbers 9+6i and 3-8i like you multiply binomials.

$$\frac{9\times 3+9\times \left(-8i\right)+6i\times 3+6\left(-8\right)i^{2}}{73}$$

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

$$\frac{9\times 3+9\times \left(-8i\right)+6i\times 3+6\left(-8\right)\left(-1\right)}{73}$$

Do the multiplications in 9\times 3+9\times \left(-8i\right)+6i\times 3+6\left(-8\right)\left(-1\right).

$$\frac{27-72i+18i+48}{73}$$

Combine the real and imaginary parts in 27-72i+18i+48.

$$\frac{27+48+\left(-72+18\right)i}{73}$$

Do the additions in 27+48+\left(-72+18\right)i.

$$\frac{75-54i}{73}$$

Divide 75-54i by 73 to get \frac{75}{73}-\frac{54}{73}i.

$$\frac{75}{73}-\frac{54}{73}i$$

Real Part

$$\frac{75}{73} = 1\frac{2}{73} = 1.0273972602739727$$

Solution Steps

Multiply both numerator and denominator of \frac{9+6i}{3+8i} by the complex conjugate of the denominator, 3-8i.

$$Re(\frac{\left(9+6i\right)\left(3-8i\right)}{\left(3+8i\right)\left(3-8i\right)})$$

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

$$Re(\frac{\left(9+6i\right)\left(3-8i\right)}{3^{2}-8^{2}i^{2}})$$

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

$$Re(\frac{\left(9+6i\right)\left(3-8i\right)}{73})$$

Multiply complex numbers 9+6i and 3-8i like you multiply binomials.

$$Re(\frac{9\times 3+9\times \left(-8i\right)+6i\times 3+6\left(-8\right)i^{2}}{73})$$

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

$$Re(\frac{9\times 3+9\times \left(-8i\right)+6i\times 3+6\left(-8\right)\left(-1\right)}{73})$$

Do the multiplications in 9\times 3+9\times \left(-8i\right)+6i\times 3+6\left(-8\right)\left(-1\right).

$$Re(\frac{27-72i+18i+48}{73})$$

Combine the real and imaginary parts in 27-72i+18i+48.

$$Re(\frac{27+48+\left(-72+18\right)i}{73})$$

Do the additions in 27+48+\left(-72+18\right)i.

$$Re(\frac{75-54i}{73})$$

Divide 75-54i by 73 to get \frac{75}{73}-\frac{54}{73}i.

$$Re(\frac{75}{73}-\frac{54}{73}i)$$

The real part of \frac{75}{73}-\frac{54}{73}i is \frac{75}{73}.

$$\frac{75}{73}$$