To multiply and simplify the complex numbers [tex]\((-2 + 2i)\)[/tex] and [tex]\((5 + 5i)\)[/tex], follow these detailed steps:
1. Express each complex number:
- [tex]\(a = -2 + 2i\)[/tex]
- [tex]\(b = 5 + 5i\)[/tex]
2. Multiply the complex numbers using the distributive property (also known as the FOIL method for this case):
[tex]\[
(-2 + 2i) \cdot (5 + 5i)
\][/tex]
3. Apply the distributive property:
[tex]\[
(-2 + 2i) \cdot (5 + 5i) = (-2) \cdot (5) + (-2) \cdot (5i) + (2i) \cdot (5) + (2i) \cdot (5i)
\][/tex]
This expands to:
[tex]\[
-10 - 10i + 10i + 10i^2
\][/tex]
4. Combine the like terms:
[tex]\[
-10 + (-10i + 10i) + 10i^2
\][/tex]
Notice that [tex]\(-10i\)[/tex] and [tex]\(10i\)[/tex] cancel each other out:
[tex]\[
-10 + 10i^2
\][/tex]
5. Recall the property of [tex]\(i^2\)[/tex]:
[tex]\[
i^2 = -1
\][/tex]
Substitute [tex]\(-1\)[/tex] for [tex]\(i^2\)[/tex] in the expression:
[tex]\[
-10 + 10(-1) = -10 - 10
\][/tex]
6. Simplify the expression:
[tex]\[
-10 - 10 = -20
\][/tex]
7. Express the final product as a complex number:
[tex]\[
-20 + 0i = -20
\][/tex]
Thus, the product of [tex]\((-2 + 2i)\)[/tex] and [tex]\((5 + 5i)\)[/tex] is [tex]\(-20\)[/tex]. Hence, the simplified result is:
[tex]\[
(-20 + 0i)
\][/tex]
Therefore, the final answer is:
[tex]\[
((-20 + 0i), (-20 + 0i))
\][/tex]
