Solve 4+5i/2-7i

$$\frac { 4 + 5 i } { 2 - 7 i }$$

Evaluate

$$-\frac{27}{53}+\frac{38}{53}i\approx -0.509433962+0.716981132i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2+7i.

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

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

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

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

$$\frac{\left(4+5i\right)\left(2+7i\right)}{53}$$

Multiply complex numbers 4+5i and 2+7i like you multiply binomials.

$$\frac{4\times 2+4\times \left(7i\right)+5i\times 2+5\times 7i^{2}}{53}$$

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

$$\frac{4\times 2+4\times \left(7i\right)+5i\times 2+5\times 7\left(-1\right)}{53}$$

Do the multiplications in 4\times 2+4\times \left(7i\right)+5i\times 2+5\times 7\left(-1\right).

$$\frac{8+28i+10i-35}{53}$$

Combine the real and imaginary parts in 8+28i+10i-35.

$$\frac{8-35+\left(28+10\right)i}{53}$$

Do the additions in 8-35+\left(28+10\right)i.

$$\frac{-27+38i}{53}$$

Divide -27+38i by 53 to get -\frac{27}{53}+\frac{38}{53}i.

$$-\frac{27}{53}+\frac{38}{53}i$$

Real Part

$$-\frac{27}{53} = -0.5094339622641509$$

Solution Steps

Multiply both numerator and denominator of \frac{4+5i}{2-7i} by the complex conjugate of the denominator, 2+7i.

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

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

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

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

$$Re(\frac{\left(4+5i\right)\left(2+7i\right)}{53})$$

Multiply complex numbers 4+5i and 2+7i like you multiply binomials.

$$Re(\frac{4\times 2+4\times \left(7i\right)+5i\times 2+5\times 7i^{2}}{53})$$

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

$$Re(\frac{4\times 2+4\times \left(7i\right)+5i\times 2+5\times 7\left(-1\right)}{53})$$

Do the multiplications in 4\times 2+4\times \left(7i\right)+5i\times 2+5\times 7\left(-1\right).

$$Re(\frac{8+28i+10i-35}{53})$$

Combine the real and imaginary parts in 8+28i+10i-35.

$$Re(\frac{8-35+\left(28+10\right)i}{53})$$

Do the additions in 8-35+\left(28+10\right)i.

$$Re(\frac{-27+38i}{53})$$

Divide -27+38i by 53 to get -\frac{27}{53}+\frac{38}{53}i.

$$Re(-\frac{27}{53}+\frac{38}{53}i)$$

The real part of -\frac{27}{53}+\frac{38}{53}i is -\frac{27}{53}.

$$-\frac{27}{53}$$