Explanation:
We have the expression:
[tex]i^3[/tex]
where i represents the complex number i defined as follows:
[tex]i=\sqrt{-1}[/tex]
To find if i^3 is real or complex, we represent it as follows:
[tex]i^3=i^2\times i[/tex]
And we find the value of i^2 using the definition of i:
[tex]i^2=(\sqrt{-1})^2[/tex]
Since the square root and the power of 2 cancel each other
[tex]\imaginaryI^2=-1[/tex]
And therefore, using this value for i^2, we can now write i^3 as follows:
[tex]\begin{gathered} \imaginaryI^3=\imaginaryI^2\times\imaginaryI \\ \downarrow \\ \imaginaryI^3=(-1)\times\imaginaryI \end{gathered}[/tex]
This simplifies to -i
[tex]\imaginaryI^3=-\imaginaryI^[/tex]
Because -i is still a complex number, that means that i^3 is a complex number.
Answer: Complex
