Question

simplify the expression to a + bi form: (4 + 5i)(12 - 10i)

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
$$(4 + 5i)(12 - 10i)=4\times12+4\times(-10i)+5i\times12+5i\times(-10i)$$

Step2: Simplify each term

Calculate the products:
$$= 48-40i + 60i-50i^{2}$$
Recall that \(i^{2}=- 1\), so substitute \(i^{2}\) with \(-1\):
$$=48-40i + 60i-50\times(-1)$$
$$=48-40i + 60i + 50$$

Step3: Combine like terms

Combine the real parts and the imaginary parts separately:
Real parts: \(48 + 50=98\)
Imaginary parts: \(-40i+60i = 20i\)
So the expression becomes \(98 + 20i\)

Answer:

\(98 + 20i\)