Which expression is equivalent to i²³³ 1 1 i i?

To simplify the expression i²³³ 1 1 i i, we start by recalling the properties of the imaginary unit i, where i = √(-1). The powers of i cycle every four terms:

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

To evaluate i²³³, we can determine its equivalent by calculating the exponent modulo 4:

233 mod 4 = 1

This indicates that i²³³ is equivalent to i¹, which is simply i.

Now we combine the expression:

i 1 1 i i

Arranging this gives us:

i + 1 + 1 + i + i

This can be simplified further:

(i + i + i) + (1 + 1) = 3i + 2

Thus, the expression i²³³ 1 1 i i is equivalent to 2 + 3i.