Solve (-8+9i)*(1/2-2/3i)

$$( - 8 + 9 i ) \cdot ( \frac { 1 } { 2 } - \frac { 2 } { 3 } i )$$

Evaluate

$$2+\frac{59}{6}i\approx 2+9.833333333i$$

Solution Steps

Multiply complex numbers -8+9i and \frac{1}{2}-\frac{2}{3}i like you multiply binomials.

$$-8\times \frac{1}{2}-8\times \left(-\frac{2}{3}i\right)+9i\times \frac{1}{2}+9\left(-\frac{2}{3}\right)i^{2}$$

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

$$-8\times \frac{1}{2}-8\times \left(-\frac{2}{3}i\right)+9i\times \frac{1}{2}+9\left(-\frac{2}{3}\right)\left(-1\right)$$

Do the multiplications.

$$-4+\frac{16}{3}i+\frac{9}{2}i+6$$

Combine the real and imaginary parts.

$$-4+6+\left(\frac{16}{3}+\frac{9}{2}\right)i$$

Do the additions.

$$2+\frac{59}{6}i$$

Real Part

$$2$$

Solution Steps

Multiply complex numbers -8+9i and \frac{1}{2}-\frac{2}{3}i like you multiply binomials.

$$Re(-8\times \frac{1}{2}-8\times \left(-\frac{2}{3}i\right)+9i\times \frac{1}{2}+9\left(-\frac{2}{3}\right)i^{2})$$

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

$$Re(-8\times \frac{1}{2}-8\times \left(-\frac{2}{3}i\right)+9i\times \frac{1}{2}+9\left(-\frac{2}{3}\right)\left(-1\right))$$

Do the multiplications in -8\times \frac{1}{2}-8\times \left(-\frac{2}{3}i\right)+9i\times \frac{1}{2}+9\left(-\frac{2}{3}\right)\left(-1\right).

$$Re(-4+\frac{16}{3}i+\frac{9}{2}i+6)$$

Combine the real and imaginary parts in -4+\frac{16}{3}i+\frac{9}{2}i+6.

$$Re(-4+6+\left(\frac{16}{3}+\frac{9}{2}\right)i)$$

Do the additions in -4+6+\left(\frac{16}{3}+\frac{9}{2}\right)i.

$$Re(2+\frac{59}{6}i)$$

The real part of 2+\frac{59}{6}i is 2.

$$2$$