Solve 5-i/7+2i

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

Evaluate

$$\frac{33}{53}-\frac{17}{53}i\approx 0.622641509-0.320754717i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{35-10i-7i-2}{53}$$

Combine the real and imaginary parts in 35-10i-7i-2.

$$\frac{35-2+\left(-10-7\right)i}{53}$$

Do the additions in 35-2+\left(-10-7\right)i.

$$\frac{33-17i}{53}$$

Divide 33-17i by 53 to get \frac{33}{53}-\frac{17}{53}i.

$$\frac{33}{53}-\frac{17}{53}i$$

Real Part

$$\frac{33}{53} = 0.6226415094339622$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{35-10i-7i-2}{53})$$

Combine the real and imaginary parts in 35-10i-7i-2.

$$Re(\frac{35-2+\left(-10-7\right)i}{53})$$

Do the additions in 35-2+\left(-10-7\right)i.

$$Re(\frac{33-17i}{53})$$

Divide 33-17i by 53 to get \frac{33}{53}-\frac{17}{53}i.

$$Re(\frac{33}{53}-\frac{17}{53}i)$$

The real part of \frac{33}{53}-\frac{17}{53}i is \frac{33}{53}.

$$\frac{33}{53}$$