To solve [tex]\((0,2)^2\)[/tex], we first represent the complex number [tex]\(0 + 2i\)[/tex]:
1. The complex number in question is [tex]\(0 + 2i\)[/tex].
Next, we need to square this complex number:
2. Recall that to square a complex number [tex]\(a + bi\)[/tex], we use the formula:
[tex]\[
(a + bi)^2 = a^2 + 2ab i + (bi)^2
\][/tex]
3. Substitute [tex]\(a = 0\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
(0 + 2i)^2 = 0^2 + 2 \cdot 0 \cdot 2i + (2i)^2
\][/tex]
4. Simplify each term:
[tex]\[
0^2 = 0
\][/tex]
[tex]\[
2 \cdot 0 \cdot 2i = 0
\][/tex]
[tex]\[
(2i)^2 = 4i^2
\][/tex]
5. Recall that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
4i^2 = 4 \cdot (-1) = -4
\][/tex]
6. Combine all terms:
[tex]\[
0 + 0 + (-4) = -4
\][/tex]
Thus, the result of squaring the complex number [tex]\(0 + 2i\)[/tex] is [tex]\(-4\)[/tex]. Therefore,
[tex]\[
(0, 2)^2 = -4.
\][/tex]
