Solve (4-6i)(10+6i)/(4-6i)+(10+6i)

$$\frac { ( 4 - 6 i ) ( 10 + 6 i ) } { ( 4 - 6 i ) + ( 10 + 6 i ) }$$

Evaluate

$$\frac{38}{7}-\frac{18}{7}i\approx 5.428571429-2.571428571i$$

Solution Steps

Multiply complex numbers 4-6i and 10+6i like you multiply binomials.

$$\frac{4\times 10+4\times \left(6i\right)-6i\times 10-6\times 6i^{2}}{4-6i+\left(10+6i\right)}$$

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

$$\frac{4\times 10+4\times \left(6i\right)-6i\times 10-6\times 6\left(-1\right)}{4-6i+\left(10+6i\right)}$$

Do the multiplications in 4\times 10+4\times \left(6i\right)-6i\times 10-6\times 6\left(-1\right).

$$\frac{40+24i-60i+36}{4-6i+\left(10+6i\right)}$$

Combine the real and imaginary parts in 40+24i-60i+36.

$$\frac{40+36+\left(24-60\right)i}{4-6i+\left(10+6i\right)}$$

Do the additions in 40+36+\left(24-60\right)i.

$$\frac{76-36i}{4-6i+\left(10+6i\right)}$$

Combine the real and imaginary parts in numbers 4-6i and 10+6i.

$$\frac{76-36i}{4+10+\left(-6+6\right)i}$$

Add 4 to 10. Add -6 to 6.

$$\frac{76-36i}{14}$$

Divide 76-36i by 14 to get \frac{38}{7}-\frac{18}{7}i.

$$\frac{38}{7}-\frac{18}{7}i$$

Real Part

$$\frac{38}{7} = 5\frac{3}{7} = 5.428571428571429$$

Solution Steps

Multiply complex numbers 4-6i and 10+6i like you multiply binomials.

$$Re(\frac{4\times 10+4\times \left(6i\right)-6i\times 10-6\times 6i^{2}}{4-6i+\left(10+6i\right)})$$

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

$$Re(\frac{4\times 10+4\times \left(6i\right)-6i\times 10-6\times 6\left(-1\right)}{4-6i+\left(10+6i\right)})$$

Do the multiplications in 4\times 10+4\times \left(6i\right)-6i\times 10-6\times 6\left(-1\right).

$$Re(\frac{40+24i-60i+36}{4-6i+\left(10+6i\right)})$$

Combine the real and imaginary parts in 40+24i-60i+36.

$$Re(\frac{40+36+\left(24-60\right)i}{4-6i+\left(10+6i\right)})$$

Do the additions in 40+36+\left(24-60\right)i.

$$Re(\frac{76-36i}{4-6i+\left(10+6i\right)})$$

Combine the real and imaginary parts in numbers 4-6i and 10+6i.

$$Re(\frac{76-36i}{4+10+\left(-6+6\right)i})$$

Add 4 to 10. Add -6 to 6.

$$Re(\frac{76-36i}{14})$$

Divide 76-36i by 14 to get \frac{38}{7}-\frac{18}{7}i.

$$Re(\frac{38}{7}-\frac{18}{7}i)$$

The real part of \frac{38}{7}-\frac{18}{7}i is \frac{38}{7}.

$$\frac{38}{7}$$