Calculate: (5+6 i) (7-8 i)

Expression: $$( 5 + 6 i ) ( 7 - 8 i )$$

Multiply complex numbers $5+6i$ and $7-8i$ like you multiply binomials.

$$5\times 7+5\times \left(-8i\right)+6i\times 7+6\left(-8\right)i^{2}$$

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

$$5\times 7+5\times \left(-8i\right)+6i\times 7+6\left(-8\right)\left(-1\right)$$

Do the multiplications.

$$35-40i+42i+48$$

Combine the real and imaginary parts.

$$35+48+\left(-40+42\right)i$$

Do the additions.

$$83+2i$$