Step by step
The question
Question: simplify i¹⁰.
i10
No real number times itself is negative.
So we use a new number: i, the square root of minus 1.
So i squared is minus 1.
i = √(−1)
i² = −1
1 Make pairs of i
10 i's make 5 pairs.
Each pair is i².
= (i2)5
2 Change i² to −1
Every i² is minus 1.
= (−1)5
Minus 1 times itself 5 times: 5 is odd, so it stays minus.
It is minus 1.
(−1)^5: 5 minus signs → −1
So i¹⁰ is −1.
= −1
3 Check
Let's check with the pattern: i, minus 1, minus i, 1, then again.
It repeats every 4.
i, i², i³, ...: i, −1, −i, 1, i, −1, −i, 1, ...
10 is 8 plus 2.
Place 2 in the pattern is −1.
The same!
10 = 2 × 4 + 2 → −1
Let's look back.
First: make pairs of i.
Then: change i² to −1.
Last: check.
Answer
= −1