Solve {l}{(4+5i)}{*(5+4i)}

$$\left. \begin{array} { l } { ( 4 + 5 i ) } \\ { \times ( 5 + 4 i ) } \end{array} \right.$$

Evaluate

$$41i$$

Solution Steps

Multiply complex numbers 4+5i and 5+4i like you multiply binomials.

$$4\times 5+4\times \left(4i\right)+5i\times 5+5\times 4i^{2}$$

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

$$4\times 5+4\times \left(4i\right)+5i\times 5+5\times 4\left(-1\right)$$

Do the multiplications.

$$20+16i+25i-20$$

Combine the real and imaginary parts.

$$20-20+\left(16+25\right)i$$

Do the additions.

$$41i$$

Real Part

$$0$$

Solution Steps

Multiply complex numbers 4+5i and 5+4i like you multiply binomials.

$$Re(4\times 5+4\times \left(4i\right)+5i\times 5+5\times 4i^{2})$$

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

$$Re(4\times 5+4\times \left(4i\right)+5i\times 5+5\times 4\left(-1\right))$$

Do the multiplications in 4\times 5+4\times \left(4i\right)+5i\times 5+5\times 4\left(-1\right).

$$Re(20+16i+25i-20)$$

Combine the real and imaginary parts in 20+16i+25i-20.

$$Re(20-20+\left(16+25\right)i)$$

Do the additions in 20-20+\left(16+25\right)i.

$$Re(41i)$$

The real part of 41i is 0.

$$0$$