Solve 2+3i/5-7i

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

Evaluate

$$-\frac{11}{74}+\frac{29}{74}i\approx -0.148648649+0.391891892i$$

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.

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

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

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

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

$$\frac{-11+29i}{74}$$

Divide -11+29i by 74 to get -\frac{11}{74}+\frac{29}{74}i.

$$-\frac{11}{74}+\frac{29}{74}i$$

Real Part

$$-\frac{11}{74} = -0.14864864864864866$$

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.

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

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

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

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

$$Re(\frac{-11+29i}{74})$$

Divide -11+29i by 74 to get -\frac{11}{74}+\frac{29}{74}i.

$$Re(-\frac{11}{74}+\frac{29}{74}i)$$

The real part of -\frac{11}{74}+\frac{29}{74}i is -\frac{11}{74}.

$$-\frac{11}{74}$$