Solve (-1/5-10i)(-5/2501+250/2501i)

$$( - \frac { 1 } { 5 } - 10 i ) ( - \frac { 5 } { 2501 } + \frac { 250 } { 2501 } i )$$

Evaluate

$$1$$

Solution Steps

Multiply complex numbers -\frac{1}{5}-10i and -\frac{5}{2501}+\frac{250}{2501}i like you multiply binomials.

$$-\frac{1}{5}\left(-\frac{5}{2501}\right)-\frac{1}{5}\times \left(\frac{250}{2501}i\right)-10i\left(-\frac{5}{2501}\right)-10\times \frac{250}{2501}i^{2}$$

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

$$-\frac{1}{5}\left(-\frac{5}{2501}\right)-\frac{1}{5}\times \left(\frac{250}{2501}i\right)-10i\left(-\frac{5}{2501}\right)-10\times \frac{250}{2501}\left(-1\right)$$

Do the multiplications.

$$\frac{1}{2501}-\frac{50}{2501}i+\frac{50}{2501}i+\frac{2500}{2501}$$

Combine the real and imaginary parts.

$$\frac{1}{2501}+\frac{2500}{2501}+\left(-\frac{50}{2501}+\frac{50}{2501}\right)i$$

Do the additions.

$$1$$

Real Part

$$1$$

Solution Steps

Multiply complex numbers -\frac{1}{5}-10i and -\frac{5}{2501}+\frac{250}{2501}i like you multiply binomials.

$$Re(-\frac{1}{5}\left(-\frac{5}{2501}\right)-\frac{1}{5}\times \left(\frac{250}{2501}i\right)-10i\left(-\frac{5}{2501}\right)-10\times \frac{250}{2501}i^{2})$$

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

$$Re(-\frac{1}{5}\left(-\frac{5}{2501}\right)-\frac{1}{5}\times \left(\frac{250}{2501}i\right)-10i\left(-\frac{5}{2501}\right)-10\times \frac{250}{2501}\left(-1\right))$$

Do the multiplications in -\frac{1}{5}\left(-\frac{5}{2501}\right)-\frac{1}{5}\times \left(\frac{250}{2501}i\right)-10i\left(-\frac{5}{2501}\right)-10\times \frac{250}{2501}\left(-1\right).

$$Re(\frac{1}{2501}-\frac{50}{2501}i+\frac{50}{2501}i+\frac{2500}{2501})$$

Combine the real and imaginary parts in \frac{1}{2501}-\frac{50}{2501}i+\frac{50}{2501}i+\frac{2500}{2501}.

$$Re(\frac{1}{2501}+\frac{2500}{2501}+\left(-\frac{50}{2501}+\frac{50}{2501}\right)i)$$

Do the additions in \frac{1}{2501}+\frac{2500}{2501}+\left(-\frac{50}{2501}+\frac{50}{2501}\right)i.

$$Re(1)$$

The real part of 1 is 1.

$$1$$