Solve 2-i/7+3i

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

Evaluate

$$\frac{11}{58}-\frac{13}{58}i\approx 0.189655172-0.224137931i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-3i.

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

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

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

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

$$\frac{\left(2-i\right)\left(7-3i\right)}{58}$$

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

$$\frac{2\times 7+2\times \left(-3i\right)-i\times 7-\left(-3i^{2}\right)}{58}$$

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

$$\frac{2\times 7+2\times \left(-3i\right)-i\times 7-\left(-3\left(-1\right)\right)}{58}$$

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

$$\frac{14-6i-7i-3}{58}$$

Combine the real and imaginary parts in 14-6i-7i-3.

$$\frac{14-3+\left(-6-7\right)i}{58}$$

Do the additions in 14-3+\left(-6-7\right)i.

$$\frac{11-13i}{58}$$

Divide 11-13i by 58 to get \frac{11}{58}-\frac{13}{58}i.

$$\frac{11}{58}-\frac{13}{58}i$$

Real Part

$$\frac{11}{58} = 0.1896551724137931$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(2-i\right)\left(7-3i\right)}{58})$$

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

$$Re(\frac{2\times 7+2\times \left(-3i\right)-i\times 7-\left(-3i^{2}\right)}{58})$$

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

$$Re(\frac{2\times 7+2\times \left(-3i\right)-i\times 7-\left(-3\left(-1\right)\right)}{58})$$

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

$$Re(\frac{14-6i-7i-3}{58})$$

Combine the real and imaginary parts in 14-6i-7i-3.

$$Re(\frac{14-3+\left(-6-7\right)i}{58})$$

Do the additions in 14-3+\left(-6-7\right)i.

$$Re(\frac{11-13i}{58})$$

Divide 11-13i by 58 to get \frac{11}{58}-\frac{13}{58}i.

$$Re(\frac{11}{58}-\frac{13}{58}i)$$

The real part of \frac{11}{58}-\frac{13}{58}i is \frac{11}{58}.

$$\frac{11}{58}$$