Question

multiply and simplify the following complex numbers: $(-1 - 5i) cdot (1 - 2i)$

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
$$(-1 - 5i) \cdot (1 - 2i) = -1 \cdot 1 + (-1) \cdot (-2i) + (-5i) \cdot 1 + (-5i) \cdot (-2i)$$

Step2: Simplify each product

Simplify each of the four products:
$$-1 \cdot 1 = -1$$
$$(-1) \cdot (-2i) = 2i$$
$$(-5i) \cdot 1 = -5i$$
$$(-5i) \cdot (-2i) = 10i^2$$
So now we have:
$$-1 + 2i - 5i + 10i^2$$

Step3: Combine like terms and use \(i^2 = -1\)

Combine the imaginary terms (\(2i - 5i = -3i\)) and substitute \(i^2 = -1\) into the last term:
$$10i^2 = 10(-1) = -10$$
Now the expression becomes:
$$-1 - 3i - 10$$

Step4: Combine the real terms

Combine the real terms (\(-1 - 10 = -11\)):
$$-11 - 3i$$

Answer:

\(-11 - 3i\)