Which expression is equivalent to i 233?
a)1
b)–1
c)i
d)–i
We have to find the value of the expression \(i^{233}\)
We know that the below values.
\(i^2=-1\\i^4=1\)
Hence, in order to find the value of the given expression, we can first rewrite it in terms of \(i^4\)
\(i^{233}=(i^4)^{58}\cdot i\)
Now, we know that \(i^4=1\)
Hence, we have
\(i^{233}=(1)^{58}\cdot i\)
\(i^{233}=1\cdot i\)
\(i^{233}=i\)
C is the correct option.
The value of \(i ^{233}\) is \(i\).
Given to us
We know the value of iota(i ) is equal to √-1.
therefore, the expression can be written as,
\(i^{233} = (\sqrt{-1})^{233}\)
It can be furthered written as,
\(i^{233} = (\sqrt{-1})^{233}\\\\\),
Using the exponential rule \(x^a \times x^b = x^{(a+b)}\)
\(i^{233} = (\sqrt{-1})^{232} \times (\sqrt{-1})\\\\\)
Using \((x^a)^b =x^{ab}\),
\(i^{233} = [(\sqrt{-1})^2]^{116} \times (\sqrt{-1})\\\\i^{233} = (-1})^{116} \times (\sqrt{-1})\\\\i^{233} = 1 \times (\sqrt{-1})\\\\i^{233} = 1 \times i\\i^{233} = i\)
Hence, the value of \(i ^{233}\) is \(i\).
Learn more about the Exponential rule:
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