Question

The tiles to the correct boxes to complete the pairs.

Match each power of i to its equivalent expression.
Tiles
1
Pairs
181
¡52
199
¡66
-1
11
-1

Answer 1

Explanation:

just keep in mind :

i = sqrt(-1)

that is all we need to know.

from there we know the forever repeating group of 4 "power of i" values :

i¹ = i

i² = sqrt(-1)² = -1

i³ = i²×i = -1×i = -i

i⁴ = i²×i² = -1×-1 = 1

and this repeats now over and over again with the following powers :

i⁵ = i⁴×i = 1×i = i

i⁶ = i⁴×i² = 1×-1 = -1

i⁷ = i⁴×i³ = 1×-i = -i

i⁸ = i⁴×i⁴ = 1×1 = 1

and so on and so on ...

so,

i⁸¹ = i⁸⁰×i = (i⁴)²⁰×i = 1²⁰×i = 1×i = i

i⁵² = (i⁴)¹³ = 1¹³ = 1

i⁹⁹ = i⁹⁶×i³ = (i⁴)²⁴×i³ = 1²⁴×-i = 1×-i = -i

i⁶⁶ = i⁶⁴×i² = (i⁴)¹⁶×i² = 1¹⁶×-1 = 1×-1 = -1

for such questions always look for the modulo (remainder) result when dividing the exponent by 4. and then use the first list or table above.