Solve 3-3i/3-2i

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

Evaluate

$$\frac{15}{13}-\frac{3}{13}i\approx 1.153846154-0.230769231i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+2i.

$$\frac{\left(3-3i\right)\left(3+2i\right)}{\left(3-2i\right)\left(3+2i\right)}$$

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

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

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

$$\frac{\left(3-3i\right)\left(3+2i\right)}{13}$$

Multiply complex numbers 3-3i and 3+2i like you multiply binomials.

$$\frac{3\times 3+3\times \left(2i\right)-3i\times 3-3\times 2i^{2}}{13}$$

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

$$\frac{3\times 3+3\times \left(2i\right)-3i\times 3-3\times 2\left(-1\right)}{13}$$

Do the multiplications in 3\times 3+3\times \left(2i\right)-3i\times 3-3\times 2\left(-1\right).

$$\frac{9+6i-9i+6}{13}$$

Combine the real and imaginary parts in 9+6i-9i+6.

$$\frac{9+6+\left(6-9\right)i}{13}$$

Do the additions in 9+6+\left(6-9\right)i.

$$\frac{15-3i}{13}$$

Divide 15-3i by 13 to get \frac{15}{13}-\frac{3}{13}i.

$$\frac{15}{13}-\frac{3}{13}i$$

Real Part

$$\frac{15}{13} = 1\frac{2}{13} = 1.1538461538461537$$

Solution Steps

Multiply both numerator and denominator of \frac{3-3i}{3-2i} by the complex conjugate of the denominator, 3+2i.

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

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

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

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

$$Re(\frac{\left(3-3i\right)\left(3+2i\right)}{13})$$

Multiply complex numbers 3-3i and 3+2i like you multiply binomials.

$$Re(\frac{3\times 3+3\times \left(2i\right)-3i\times 3-3\times 2i^{2}}{13})$$

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

$$Re(\frac{3\times 3+3\times \left(2i\right)-3i\times 3-3\times 2\left(-1\right)}{13})$$

Do the multiplications in 3\times 3+3\times \left(2i\right)-3i\times 3-3\times 2\left(-1\right).

$$Re(\frac{9+6i-9i+6}{13})$$

Combine the real and imaginary parts in 9+6i-9i+6.

$$Re(\frac{9+6+\left(6-9\right)i}{13})$$

Do the additions in 9+6+\left(6-9\right)i.

$$Re(\frac{15-3i}{13})$$

Divide 15-3i by 13 to get \frac{15}{13}-\frac{3}{13}i.

$$Re(\frac{15}{13}-\frac{3}{13}i)$$

The real part of \frac{15}{13}-\frac{3}{13}i is \frac{15}{13}.

$$\frac{15}{13}$$