Solve 9-9i/7+4i

$$\frac { 9 - 9 i } { 7 + 4 i }$$

Evaluate

$$\frac{27}{65}-\frac{99}{65}i\approx 0.415384615-1.523076923i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-4i.

$$\frac{\left(9-9i\right)\left(7-4i\right)}{\left(7+4i\right)\left(7-4i\right)}$$

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

$$\frac{\left(9-9i\right)\left(7-4i\right)}{7^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(9-9i\right)\left(7-4i\right)}{65}$$

Multiply complex numbers 9-9i and 7-4i like you multiply binomials.

$$\frac{9\times 7+9\times \left(-4i\right)-9i\times 7-9\left(-4\right)i^{2}}{65}$$

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

$$\frac{9\times 7+9\times \left(-4i\right)-9i\times 7-9\left(-4\right)\left(-1\right)}{65}$$

Do the multiplications in 9\times 7+9\times \left(-4i\right)-9i\times 7-9\left(-4\right)\left(-1\right).

$$\frac{63-36i-63i-36}{65}$$

Combine the real and imaginary parts in 63-36i-63i-36.

$$\frac{63-36+\left(-36-63\right)i}{65}$$

Do the additions in 63-36+\left(-36-63\right)i.

$$\frac{27-99i}{65}$$

Divide 27-99i by 65 to get \frac{27}{65}-\frac{99}{65}i.

$$\frac{27}{65}-\frac{99}{65}i$$

Real Part

$$\frac{27}{65} = 0.4153846153846154$$

Solution Steps

Multiply both numerator and denominator of \frac{9-9i}{7+4i} by the complex conjugate of the denominator, 7-4i.

$$Re(\frac{\left(9-9i\right)\left(7-4i\right)}{\left(7+4i\right)\left(7-4i\right)})$$

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

$$Re(\frac{\left(9-9i\right)\left(7-4i\right)}{7^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(9-9i\right)\left(7-4i\right)}{65})$$

Multiply complex numbers 9-9i and 7-4i like you multiply binomials.

$$Re(\frac{9\times 7+9\times \left(-4i\right)-9i\times 7-9\left(-4\right)i^{2}}{65})$$

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

$$Re(\frac{9\times 7+9\times \left(-4i\right)-9i\times 7-9\left(-4\right)\left(-1\right)}{65})$$

Do the multiplications in 9\times 7+9\times \left(-4i\right)-9i\times 7-9\left(-4\right)\left(-1\right).

$$Re(\frac{63-36i-63i-36}{65})$$

Combine the real and imaginary parts in 63-36i-63i-36.

$$Re(\frac{63-36+\left(-36-63\right)i}{65})$$

Do the additions in 63-36+\left(-36-63\right)i.

$$Re(\frac{27-99i}{65})$$

Divide 27-99i by 65 to get \frac{27}{65}-\frac{99}{65}i.

$$Re(\frac{27}{65}-\frac{99}{65}i)$$

The real part of \frac{27}{65}-\frac{99}{65}i is \frac{27}{65}.

$$\frac{27}{65}$$