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djeljenike

Answered question

2021-08-10

To calculate: The simplified expression for the expression

i^{1000} + i^{1002}

krolaniaN

Skilled2021-08-11Added 86 answers

Formula used:
By the definition of

i = \sqrt{- 1}

, it follows that:

i^{2} = - 1

i^{3} = - i

i^{4} = 1

To simplify higher powers of i, decompose the expression into multiples of

i^{4}

and write the remaining factors as

i, i^{2}

i^{3}

.
Calculation:
Consider the provided expression:

i^{1000} + i^{1002}

Now, to simplify higher powers of i, decompose the expression into multiples of

i^{4} (i^{4} = 1)

and write the remaining factors as

i, i^{2}

i^{3}

.
Thus, for the provoded expression:

i^{1000} + i^{1002} = i^{4 (250)} + i^{4 (250)} \cdot i^{2}

= 1^{250} + 1^{250} \cdot i^{2}

Now, as

i^{2} = - 1

, thus:

i^{1000} + i^{1002} = 1^{250} + 1^{250} \cdot i^{2}

= 1 + (- 1)

= 0

Thus, the simplified expression for the expression

i^{1000} + i^{1002}

is 0.

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