Step by step
The question
Question: simplify i³.
i3
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
3 i's make 1 pairs, and one i is left.
Each pair is i².
= (i2)1 × i
2 Change i² to −1
Every i² is minus 1.
= (−1)1 × i
Minus 1 times itself 1 times: 1 is odd, so it stays minus.
It is minus 1.
(−1)^1: 1 minus signs → −1
So it is minus 1 times i.
= −1 × i
So i³ is −i.
= −i
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
3 is 0 plus 3.
Place 3 in the pattern is −i.
The same!
3 = 0 × 4 + 3 → −i
Let's look back.
First: make pairs of i.
Then: change i² to −1.
Last: check.
Answer
= −i