Solve 7+5i/1-4i

$$\frac{ 7+5i }{ 1-4i }$$

Evaluate

$$-\frac{13}{17}+\frac{33}{17}i\approx -0.764705882+1.941176471i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+4i.

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

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

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

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

$$\frac{\left(7+5i\right)\left(1+4i\right)}{17}$$

Multiply complex numbers 7+5i and 1+4i like you multiply binomials.

$$\frac{7\times 1+7\times \left(4i\right)+5i\times 1+5\times 4i^{2}}{17}$$

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

$$\frac{7\times 1+7\times \left(4i\right)+5i\times 1+5\times 4\left(-1\right)}{17}$$

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

$$\frac{7+28i+5i-20}{17}$$

Combine the real and imaginary parts in 7+28i+5i-20.

$$\frac{7-20+\left(28+5\right)i}{17}$$

Do the additions in 7-20+\left(28+5\right)i.

$$\frac{-13+33i}{17}$$

Divide -13+33i by 17 to get -\frac{13}{17}+\frac{33}{17}i.

$$-\frac{13}{17}+\frac{33}{17}i$$

Real Part

$$-\frac{13}{17} = -0.7647058823529411$$

Solution Steps

Multiply both numerator and denominator of \frac{7+5i}{1-4i} by the complex conjugate of the denominator, 1+4i.

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

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

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

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

$$Re(\frac{\left(7+5i\right)\left(1+4i\right)}{17})$$

Multiply complex numbers 7+5i and 1+4i like you multiply binomials.

$$Re(\frac{7\times 1+7\times \left(4i\right)+5i\times 1+5\times 4i^{2}}{17})$$

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

$$Re(\frac{7\times 1+7\times \left(4i\right)+5i\times 1+5\times 4\left(-1\right)}{17})$$

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

$$Re(\frac{7+28i+5i-20}{17})$$

Combine the real and imaginary parts in 7+28i+5i-20.

$$Re(\frac{7-20+\left(28+5\right)i}{17})$$

Do the additions in 7-20+\left(28+5\right)i.

$$Re(\frac{-13+33i}{17})$$

Divide -13+33i by 17 to get -\frac{13}{17}+\frac{33}{17}i.

$$Re(-\frac{13}{17}+\frac{33}{17}i)$$

The real part of -\frac{13}{17}+\frac{33}{17}i is -\frac{13}{17}.

$$-\frac{13}{17}$$