Solve 6+4i/2-3i

$$\frac { 6 + 4 i } { 2 - 3 i }$$

Evaluate

$$2i$$

Solution Steps

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

$$\frac{\left(6+4i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}$$

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

$$\frac{\left(6+4i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(6+4i\right)\left(2+3i\right)}{13}$$

Multiply complex numbers 6+4i and 2+3i like you multiply binomials.

$$\frac{6\times 2+6\times \left(3i\right)+4i\times 2+4\times 3i^{2}}{13}$$

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

$$\frac{6\times 2+6\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right)}{13}$$

Do the multiplications in 6\times 2+6\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right).

$$\frac{12+18i+8i-12}{13}$$

Combine the real and imaginary parts in 12+18i+8i-12.

$$\frac{12-12+\left(18+8\right)i}{13}$$

Do the additions in 12-12+\left(18+8\right)i.

$$\frac{26i}{13}$$

Divide 26i by 13 to get 2i.

$$2i$$

Real Part

$$0$$

Solution Steps

Multiply both numerator and denominator of \frac{6+4i}{2-3i} by the complex conjugate of the denominator, 2+3i.

$$Re(\frac{\left(6+4i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)})$$

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

$$Re(\frac{\left(6+4i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(6+4i\right)\left(2+3i\right)}{13})$$

Multiply complex numbers 6+4i and 2+3i like you multiply binomials.

$$Re(\frac{6\times 2+6\times \left(3i\right)+4i\times 2+4\times 3i^{2}}{13})$$

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

$$Re(\frac{6\times 2+6\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right)}{13})$$

Do the multiplications in 6\times 2+6\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right).

$$Re(\frac{12+18i+8i-12}{13})$$

Combine the real and imaginary parts in 12+18i+8i-12.

$$Re(\frac{12-12+\left(18+8\right)i}{13})$$

Do the additions in 12-12+\left(18+8\right)i.

$$Re(\frac{26i}{13})$$

Divide 26i by 13 to get 2i.

$$Re(2i)$$

The real part of 2i is 0.

$$0$$