Which expression is equivalent to i^(233) * (1 + i)?

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To simplify the expression i^(233) * (1 + i), let’s break it down into parts.

Understanding the Powers of i

The imaginary unit i is defined as the square root of -1. The powers of i follow a cyclic pattern:

  • i^1 = i
  • i^2 = -1
  • i^3 = -i
  • i^4 = 1

This cycle repeats every four powers. So, to find i^(233), we can find the remainder when 233 is divided by 4:

233 รท 4 = 58 remainder 1

This means that i^(233) = i^1 = i.

Substituting Back into the Expression

Now, we replace i^(233) in the original expression:

i^(233) * (1 + i) = i * (1 + i)

Distributing

Next, we distribute i:

= i * 1 + i * i = i + i^2

Since i^2 = -1, we replace i^2:

= i - 1

Final Answer

Thus, the expression equivalent to i^(233) * (1 + i) simplifies to:

i - 1

So, the final answer is i - 1.

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