Question

multiply and simplify the following complex numbers: (4 + 4i)·(-2 - 5i)

Explanation:

Step1: Apply distributive property (FOIL)

We use the distributive property (also known as the FOIL method for binomials) to multiply the two complex numbers. The formula for multiplying \((a + bi)(c + di)\) is \(ac + adi + bci + bdi^2\). Here, \(a = 4\), \(b = 4\), \(c=-2\), and \(d = - 5\). So we have:
\((4+4i)\cdot(-2 - 5i)=4\times(-2)+4\times(-5i)+4i\times(-2)+4i\times(-5i)\)

Step2: Simplify each term

• For the first term: \(4\times(-2)=-8\)
• For the second term: \(4\times(-5i)=-20i\)
• For the third term: \(4i\times(-2)=-8i\)
• For the fourth term: \(4i\times(-5i)=-20i^{2}\). Since \(i^{2}=- 1\), this term becomes \(-20\times(-1) = 20\)

Step3: Combine like terms

Now we combine the real parts and the imaginary parts separately. The real parts are \(-8\) and \(20\), and the imaginary parts are \(-20i\) and \(-8i\).
Real parts: \(-8 + 20=12\)
Imaginary parts: \(-20i-8i=-28i\)
So the result is \(12-28i\)

Answer:

\(12 - 28i\)