Solve 7+4i/5+3i

$$\frac { 7 + 4 i } { 5 + 3 i }$$

Evaluate

$$\frac{47}{34}-\frac{1}{34}i\approx 1.382352941-0.029411765i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5-3i.

$$\frac{\left(7+4i\right)\left(5-3i\right)}{\left(5+3i\right)\left(5-3i\right)}$$

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

$$\frac{\left(7+4i\right)\left(5-3i\right)}{5^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(7+4i\right)\left(5-3i\right)}{34}$$

Multiply complex numbers 7+4i and 5-3i like you multiply binomials.

$$\frac{7\times 5+7\times \left(-3i\right)+4i\times 5+4\left(-3\right)i^{2}}{34}$$

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

$$\frac{7\times 5+7\times \left(-3i\right)+4i\times 5+4\left(-3\right)\left(-1\right)}{34}$$

Do the multiplications in 7\times 5+7\times \left(-3i\right)+4i\times 5+4\left(-3\right)\left(-1\right).

$$\frac{35-21i+20i+12}{34}$$

Combine the real and imaginary parts in 35-21i+20i+12.

$$\frac{35+12+\left(-21+20\right)i}{34}$$

Do the additions in 35+12+\left(-21+20\right)i.

$$\frac{47-i}{34}$$

Divide 47-i by 34 to get \frac{47}{34}-\frac{1}{34}i.

$$\frac{47}{34}-\frac{1}{34}i$$

Real Part

$$\frac{47}{34} = 1\frac{13}{34} = 1.3823529411764706$$

Solution Steps

Multiply both numerator and denominator of \frac{7+4i}{5+3i} by the complex conjugate of the denominator, 5-3i.

$$Re(\frac{\left(7+4i\right)\left(5-3i\right)}{\left(5+3i\right)\left(5-3i\right)})$$

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

$$Re(\frac{\left(7+4i\right)\left(5-3i\right)}{5^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(7+4i\right)\left(5-3i\right)}{34})$$

Multiply complex numbers 7+4i and 5-3i like you multiply binomials.

$$Re(\frac{7\times 5+7\times \left(-3i\right)+4i\times 5+4\left(-3\right)i^{2}}{34})$$

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

$$Re(\frac{7\times 5+7\times \left(-3i\right)+4i\times 5+4\left(-3\right)\left(-1\right)}{34})$$

Do the multiplications in 7\times 5+7\times \left(-3i\right)+4i\times 5+4\left(-3\right)\left(-1\right).

$$Re(\frac{35-21i+20i+12}{34})$$

Combine the real and imaginary parts in 35-21i+20i+12.

$$Re(\frac{35+12+\left(-21+20\right)i}{34})$$

Do the additions in 35+12+\left(-21+20\right)i.

$$Re(\frac{47-i}{34})$$

Divide 47-i by 34 to get \frac{47}{34}-\frac{1}{34}i.

$$Re(\frac{47}{34}-\frac{1}{34}i)$$

The real part of \frac{47}{34}-\frac{1}{34}i is \frac{47}{34}.

$$\frac{47}{34}$$