Step by step
The question
Question: simplify i¹⁴.
The numbers we know are called real numbers.
No real number times itself is negative.
3 times 3 is 9, and minus 3 times minus 3 is 9 too.
So we use a new number: i, the square root of minus 1.
A square root times itself gives back what is under it.
So i squared is minus 1.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Make pairs of i
14 i's make 7 pairs.
Each pair is i².
2 Change i² to −1
Every i² is minus 1.
Minus 1 times itself 7 times: 7 is odd, so it stays minus.
It is minus 1.
So i¹⁴ is −1.
3 Check
Let's check with the pattern: i, minus 1, minus i, 1, then again.
It repeats every 4.
14 is 12 plus 2.
Place 2 in the pattern is −1.
The same!
4 Pick the choice
Our answer is −1.
That is choice A.
5 Why not the others
B is −i.
Wrong: that is i³: 14 ÷ 4 leaves 2, so it is like i².
C is 1.
Wrong: that is i⁴: 14 is not a multiple of 4.
D is i.
Wrong: that is i¹: 14 ÷ 4 leaves 2, so it is like i².
Let's look back.
First: make pairs of i.
Then: change i² to −1.
Then: check.
Then: pick the choice.
Last: why not the others.