To simplify the expression i^71, we need to understand the properties of the imaginary unit i. The imaginary unit i is defined as the square root of -1, which means i^2 is equal to -1.
We can find a pattern in the powers of i:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
From this pattern, we can observe that the powers of i repeat every four powers. So, i^5 will be the same as i^1, i^6 will be the same as i^2, and so on.
Now, let's simplify i^71 using this pattern:
i^71 = i^(4 * 17 + 3)
Since we know that i^4 is equal to 1, we can rewrite i^71 as:
i^71 = (i^4)^17 * i^3
Now, we substitute the value of i^4 as 1:
i^71 = 1^17 * i^3
We can simplify further:
i^71 = 1 * i^3
Using the pattern of i, we know that i^3 is equal to -i:
i^71 = -i
Therefore, the simplified form of i^71 is -i.