Final answer:
To find the answer to i⁴⁵, we need to understand the properties of imaginary numbers. Imaginary numbers are represented by the letter i, where i = √-1. The power of i follows a pattern, where i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. After that, the pattern repeats itself. Therefore, i⁴⁵ can be broken down into i⁴⁵ = (i⁴)¹¹ + 1 = 1¹¹ + 1 = 1 + 1 = 2. However, since the power of i is always in multiples of 4, we can simplify this even further. i⁴⁵ = (i⁴)¹¹ + 1 = (1)¹¹ + 1 = 1 + 1 = 2, thus the correct option is option is C.
Explanation:
To find the answer to i⁴⁵, we need to understand the properties of imaginary numbers. Imaginary numbers are represented by the letter i, where i = √-1. The power of i follows a pattern, where i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. After that, the pattern repeats itself. Therefore, i⁴⁵ can be broken down into i⁴⁵ = (i⁴)¹¹ + 1 = 1¹¹ + 1 = 1 + 1 = 2. However, since the power of i is always in multiples of 4, we can simplify this even further. i⁴⁵ = (i⁴)¹¹ + 1 = (1)¹¹ + 1 = 1 + 1 = 2. Hence, the final answer is C. i.
Explanation: In the first part, we broke down i⁴⁵ into i⁴⁵ = (i⁴)¹¹ + 1. This is because the power of i follows a pattern, where the power is always in multiples of 4. Therefore, we can simplify the power by dividing it by 4. In this case, 45 divided by 4 gives us a remainder of 1, hence we have (i⁴)¹¹. The power of i⁴ is always equal to 1, hence we can further simplify this to (1)¹¹. Since any number raised to the power of 1 is itself, we get (1)¹¹ = 1. Finally, we add 1 to get the final answer of 2.
In the second part, we can also use De Moivre's theorem to find the answer to i⁴⁵. According to De Moivre's theorem, (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ). In this case, n = 45, and θ = π/2. Therefore, we have (cos(π/2) + i sin(π/2))^45 = cos(45π/2) + i sin(45π/2). Simplifying this, we get (0 + i)^45 = i^45. As we know, i^45 can be simplified to (i^4)^11 + 1 = (1)^11 + 1 = 1 + 1 = 2.
Another method to find the answer to i⁴⁵ is by using Euler's formula, e^(iθ) = cosθ + i sinθ. In this case, θ = π/2. Therefore, we have e^(iπ/2) = cos(π/2) + i sin(π/2) = 0 + i. Raising this to the power of 45, we get (e^(iπ/2))^45 = (0 + i)^45 = i^45. Again, we can simplify this using the same method as before to get the final answer of 2.
In conclusion, the answer to i⁴⁵ is equal to 2. This can be found by using the properties of imaginary numbers, De Moivre's theorem, or Euler's formula. The key is to understand the pattern of powers of i and simplify the power to make it easier to calculate. This highlights the importance of understanding the properties and patterns of numbers in solving mathematical problems.