Solve 7+4i/2-3i

$$\frac{ 7+4i }{ 2-3i }$$

Evaluate

$$\frac{2}{13}+\frac{29}{13}i\approx 0.153846154+2.230769231i$$

Solution Steps

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

$$\frac{\left(7+4i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)}$$

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

$$\frac{\left(7+4i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(7+4i\right)\left(2+3i\right)}{13}$$

Multiply complex numbers 7+4i and 2+3i like you multiply binomials.

$$\frac{7\times 2+7\times \left(3i\right)+4i\times 2+4\times 3i^{2}}{13}$$

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

$$\frac{7\times 2+7\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right)}{13}$$

Do the multiplications in 7\times 2+7\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right).

$$\frac{14+21i+8i-12}{13}$$

Combine the real and imaginary parts in 14+21i+8i-12.

$$\frac{14-12+\left(21+8\right)i}{13}$$

Do the additions in 14-12+\left(21+8\right)i.

$$\frac{2+29i}{13}$$

Divide 2+29i by 13 to get \frac{2}{13}+\frac{29}{13}i.

$$\frac{2}{13}+\frac{29}{13}i$$

Real Part

$$\frac{2}{13} = 0.15384615384615385$$

Solution Steps

Multiply both numerator and denominator of \frac{7+4i}{2-3i} by the complex conjugate of the denominator, 2+3i.

$$Re(\frac{\left(7+4i\right)\left(2+3i\right)}{\left(2-3i\right)\left(2+3i\right)})$$

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

$$Re(\frac{\left(7+4i\right)\left(2+3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(7+4i\right)\left(2+3i\right)}{13})$$

Multiply complex numbers 7+4i and 2+3i like you multiply binomials.

$$Re(\frac{7\times 2+7\times \left(3i\right)+4i\times 2+4\times 3i^{2}}{13})$$

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

$$Re(\frac{7\times 2+7\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right)}{13})$$

Do the multiplications in 7\times 2+7\times \left(3i\right)+4i\times 2+4\times 3\left(-1\right).

$$Re(\frac{14+21i+8i-12}{13})$$

Combine the real and imaginary parts in 14+21i+8i-12.

$$Re(\frac{14-12+\left(21+8\right)i}{13})$$

Do the additions in 14-12+\left(21+8\right)i.

$$Re(\frac{2+29i}{13})$$

Divide 2+29i by 13 to get \frac{2}{13}+\frac{29}{13}i.

$$Re(\frac{2}{13}+\frac{29}{13}i)$$

The real part of \frac{2}{13}+\frac{29}{13}i is \frac{2}{13}.

$$\frac{2}{13}$$