Question

To calculate: i^{1000} + i^{1002}

Factors and multiples
ANSWERED
asked 2021-08-10
To calculate: The simplified expression for the expression \(\displaystyle{i}^{{{1000}}}+{i}^{{{1002}}}\)

Answers (1)

2021-08-11
Formula used:
By the definition of \(\displaystyle{i}=\sqrt{{-{1}}}\), it follows that:
\(\displaystyle{i}^{{{2}}}=-{1}\)
\(\displaystyle{i}^{{{3}}}=-{i}\)
\(\displaystyle{i}^{{{4}}}={1}\)
To simplify higher powers of i, decompose the expression into multiples of \(\displaystyle{i}^{{{4}}}\) and write the remaining factors as \(\displaystyle{i},{i}^{{{2}}}\) or \(\displaystyle{i}^{{{3}}}\).
Calculation:
Consider the provided expression:
\(\displaystyle{i}^{{{1000}}}+{i}^{{{1002}}}\)
Now, to simplify higher powers of i, decompose the expression into multiples of \(\displaystyle{i}^{{{4}}}{\left({i}^{{{4}}}={1}\right)}\) and write the remaining factors as \(\displaystyle{i},{i}^{{{2}}}\) or \(\displaystyle{i}^{{{3}}}\).
Thus, for the provoded expression:
\(\displaystyle{i}^{{{1000}}}+{i}^{{{1002}}}={i}^{{{4}{\left({250}\right)}}}+{i}^{{{4}{\left({250}\right)}}}\cdot{i}^{{{2}}}\)
\(\displaystyle={1}^{{{250}}}+{1}^{{{250}}}\cdot{i}^{{{2}}}\)
Now, as \(\displaystyle{i}^{{{2}}}=-{1}\), thus:
\(\displaystyle{i}^{{{1000}}}+{i}^{{{1002}}}={1}^{{{250}}}+{1}^{{{250}}}\cdot{i}^{{{2}}}\)
\(\displaystyle={1}+{\left(-{1}\right)}\)
\(\displaystyle={0}\)
Thus, the simplified expression for the expression \(\displaystyle{i}^{{{1000}}}+{i}^{{{1002}}}\) is 0.
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