Solve (4+3i)/(3-2i)

$$\frac { ( 4 + 3 i ) } { ( 3 - 2 i ) }$$

Evaluate

$$\frac{6}{13}+\frac{17}{13}i\approx 0.461538462+1.307692308i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(4+3i\right)\left(3+2i\right)}{13}$$

Multiply complex numbers 4+3i and 3+2i like you multiply binomials.

$$\frac{4\times 3+4\times \left(2i\right)+3i\times 3+3\times 2i^{2}}{13}$$

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

$$\frac{4\times 3+4\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right)}{13}$$

Do the multiplications in 4\times 3+4\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right).

$$\frac{12+8i+9i-6}{13}$$

Combine the real and imaginary parts in 12+8i+9i-6.

$$\frac{12-6+\left(8+9\right)i}{13}$$

Do the additions in 12-6+\left(8+9\right)i.

$$\frac{6+17i}{13}$$

Divide 6+17i by 13 to get \frac{6}{13}+\frac{17}{13}i.

$$\frac{6}{13}+\frac{17}{13}i$$

Real Part

$$\frac{6}{13} = 0.46153846153846156$$

Solution Steps

Multiply both numerator and denominator of \frac{4+3i}{3-2i} by the complex conjugate of the denominator, 3+2i.

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

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

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

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

$$Re(\frac{\left(4+3i\right)\left(3+2i\right)}{13})$$

Multiply complex numbers 4+3i and 3+2i like you multiply binomials.

$$Re(\frac{4\times 3+4\times \left(2i\right)+3i\times 3+3\times 2i^{2}}{13})$$

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

$$Re(\frac{4\times 3+4\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right)}{13})$$

Do the multiplications in 4\times 3+4\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right).

$$Re(\frac{12+8i+9i-6}{13})$$

Combine the real and imaginary parts in 12+8i+9i-6.

$$Re(\frac{12-6+\left(8+9\right)i}{13})$$

Do the additions in 12-6+\left(8+9\right)i.

$$Re(\frac{6+17i}{13})$$

Divide 6+17i by 13 to get \frac{6}{13}+\frac{17}{13}i.

$$Re(\frac{6}{13}+\frac{17}{13}i)$$

The real part of \frac{6}{13}+\frac{17}{13}i is \frac{6}{13}.

$$\frac{6}{13}$$