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.

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.