Solve 4+2i/3+5i

$$\frac { 4 + 2 i } { 3 + 5 i }$$

Evaluate

$$\frac{11}{17}-\frac{7}{17}i\approx 0.647058824-0.411764706i$$

Solution Steps

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

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

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

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

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

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

Multiply complex numbers 4+2i and 3-5i like you multiply binomials.

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

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

$$\frac{4\times 3+4\times \left(-5i\right)+2i\times 3+2\left(-5\right)\left(-1\right)}{34}$$

Do the multiplications in 4\times 3+4\times \left(-5i\right)+2i\times 3+2\left(-5\right)\left(-1\right).

$$\frac{12-20i+6i+10}{34}$$

Combine the real and imaginary parts in 12-20i+6i+10.

$$\frac{12+10+\left(-20+6\right)i}{34}$$

Do the additions in 12+10+\left(-20+6\right)i.

$$\frac{22-14i}{34}$$

Divide 22-14i by 34 to get \frac{11}{17}-\frac{7}{17}i.

$$\frac{11}{17}-\frac{7}{17}i$$

Real Part

$$\frac{11}{17} = 0.6470588235294118$$

Solution Steps

Multiply both numerator and denominator of \frac{4+2i}{3+5i} by the complex conjugate of the denominator, 3-5i.

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

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

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

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

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

Multiply complex numbers 4+2i and 3-5i like you multiply binomials.

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

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

$$Re(\frac{4\times 3+4\times \left(-5i\right)+2i\times 3+2\left(-5\right)\left(-1\right)}{34})$$

Do the multiplications in 4\times 3+4\times \left(-5i\right)+2i\times 3+2\left(-5\right)\left(-1\right).

$$Re(\frac{12-20i+6i+10}{34})$$

Combine the real and imaginary parts in 12-20i+6i+10.

$$Re(\frac{12+10+\left(-20+6\right)i}{34})$$

Do the additions in 12+10+\left(-20+6\right)i.

$$Re(\frac{22-14i}{34})$$

Divide 22-14i by 34 to get \frac{11}{17}-\frac{7}{17}i.

$$Re(\frac{11}{17}-\frac{7}{17}i)$$

The real part of \frac{11}{17}-\frac{7}{17}i is \frac{11}{17}.

$$\frac{11}{17}$$