Solve 2+3i/5+7i

$$\frac { 2 + 3 i } { 5 + 7 i }$$

Evaluate

$$\frac{31}{74}+\frac{1}{74}i\approx 0.418918919+0.013513514i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5-7i.

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

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

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

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

$$\frac{\left(2+3i\right)\left(5-7i\right)}{74}$$

Multiply complex numbers 2+3i and 5-7i like you multiply binomials.

$$\frac{2\times 5+2\times \left(-7i\right)+3i\times 5+3\left(-7\right)i^{2}}{74}$$

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

$$\frac{2\times 5+2\times \left(-7i\right)+3i\times 5+3\left(-7\right)\left(-1\right)}{74}$$

Do the multiplications in 2\times 5+2\times \left(-7i\right)+3i\times 5+3\left(-7\right)\left(-1\right).

$$\frac{10-14i+15i+21}{74}$$

Combine the real and imaginary parts in 10-14i+15i+21.

$$\frac{10+21+\left(-14+15\right)i}{74}$$

Do the additions in 10+21+\left(-14+15\right)i.

$$\frac{31+i}{74}$$

Divide 31+i by 74 to get \frac{31}{74}+\frac{1}{74}i.

$$\frac{31}{74}+\frac{1}{74}i$$

Real Part

$$\frac{31}{74} = 0.4189189189189189$$

Solution Steps

Multiply both numerator and denominator of \frac{2+3i}{5+7i} by the complex conjugate of the denominator, 5-7i.

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

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

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

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

$$Re(\frac{\left(2+3i\right)\left(5-7i\right)}{74})$$

Multiply complex numbers 2+3i and 5-7i like you multiply binomials.

$$Re(\frac{2\times 5+2\times \left(-7i\right)+3i\times 5+3\left(-7\right)i^{2}}{74})$$

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

$$Re(\frac{2\times 5+2\times \left(-7i\right)+3i\times 5+3\left(-7\right)\left(-1\right)}{74})$$

Do the multiplications in 2\times 5+2\times \left(-7i\right)+3i\times 5+3\left(-7\right)\left(-1\right).

$$Re(\frac{10-14i+15i+21}{74})$$

Combine the real and imaginary parts in 10-14i+15i+21.

$$Re(\frac{10+21+\left(-14+15\right)i}{74})$$

Do the additions in 10+21+\left(-14+15\right)i.

$$Re(\frac{31+i}{74})$$

Divide 31+i by 74 to get \frac{31}{74}+\frac{1}{74}i.

$$Re(\frac{31}{74}+\frac{1}{74}i)$$

The real part of \frac{31}{74}+\frac{1}{74}i is \frac{31}{74}.

$$\frac{31}{74}$$