Solve 9+6i/8-9i

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

Evaluate

$$\frac{18}{145}+\frac{129}{145}i\approx 0.124137931+0.889655172i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 8+9i.

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

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

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

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

$$\frac{\left(9+6i\right)\left(8+9i\right)}{145}$$

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

$$\frac{9\times 8+9\times \left(9i\right)+6i\times 8+6\times 9i^{2}}{145}$$

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

$$\frac{9\times 8+9\times \left(9i\right)+6i\times 8+6\times 9\left(-1\right)}{145}$$

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

$$\frac{72+81i+48i-54}{145}$$

Combine the real and imaginary parts in 72+81i+48i-54.

$$\frac{72-54+\left(81+48\right)i}{145}$$

Do the additions in 72-54+\left(81+48\right)i.

$$\frac{18+129i}{145}$$

Divide 18+129i by 145 to get \frac{18}{145}+\frac{129}{145}i.

$$\frac{18}{145}+\frac{129}{145}i$$

Real Part

$$\frac{18}{145} = 0.12413793103448276$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(9+6i\right)\left(8+9i\right)}{145})$$

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

$$Re(\frac{9\times 8+9\times \left(9i\right)+6i\times 8+6\times 9i^{2}}{145})$$

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

$$Re(\frac{9\times 8+9\times \left(9i\right)+6i\times 8+6\times 9\left(-1\right)}{145})$$

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

$$Re(\frac{72+81i+48i-54}{145})$$

Combine the real and imaginary parts in 72+81i+48i-54.

$$Re(\frac{72-54+\left(81+48\right)i}{145})$$

Do the additions in 72-54+\left(81+48\right)i.

$$Re(\frac{18+129i}{145})$$

Divide 18+129i by 145 to get \frac{18}{145}+\frac{129}{145}i.

$$Re(\frac{18}{145}+\frac{129}{145}i)$$

The real part of \frac{18}{145}+\frac{129}{145}i is \frac{18}{145}.

$$\frac{18}{145}$$