The conjugate of a complex number is a complex number with the same real part, but a negated imaginary part. The conjugate of [tex]i^3 - 4[/tex] is [tex]-4 + i[/tex]
Let the complex number be N. So:
[tex]N = i^3 - 4[/tex]
In complex numbers,
[tex]i = \sqrt{-1}[/tex]
Take cube of both sides
[tex]i^3 = (\sqrt{-1})^3[/tex]
Split
[tex]i^3 = (\sqrt{-1}) \times (\sqrt{-1}) \times (\sqrt{-1})[/tex]
[tex]i^3 = -1 \times (\sqrt{-1})[/tex]
Substitute [tex]i = \sqrt{-1}[/tex]
[tex]i^3 = -1 \times i[/tex]
[tex]i^3 = -i[/tex]
So, we have:
[tex]N = i^3 - 4[/tex]
[tex]N = -i - 4[/tex]
Rewrite as:
[tex]N = -4 - i[/tex]
A complex number a + bi has a- bi as its conjugate.
So, the conjugate of N is:
[tex]N' = -4 + i[/tex]
Hence, (c) is correct
Read more about conjugates at:
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