Answer :
To simplify powers of [tex]\(i\)[/tex], first recall the definition of [tex]\(i\)[/tex]:
[tex]\[i = \sqrt{-1}\][/tex]
Given this definition, we can find the square of [tex]\(i\)[/tex]:
[tex]\[ i^2 = (\sqrt{-1})^2 \][/tex]
By the property of squares and square roots, we know:
[tex]\[ (\sqrt{-1})^2 = -1 \][/tex]
Thus, the simplified value of [tex]\(i^2\)[/tex] is:
[tex]\[ i^2 = -1 \][/tex]
Therefore, filling in the expression:
[tex]\[ i^2 = \boxed{-1} \][/tex]
[tex]\[i = \sqrt{-1}\][/tex]
Given this definition, we can find the square of [tex]\(i\)[/tex]:
[tex]\[ i^2 = (\sqrt{-1})^2 \][/tex]
By the property of squares and square roots, we know:
[tex]\[ (\sqrt{-1})^2 = -1 \][/tex]
Thus, the simplified value of [tex]\(i^2\)[/tex] is:
[tex]\[ i^2 = -1 \][/tex]
Therefore, filling in the expression:
[tex]\[ i^2 = \boxed{-1} \][/tex]
