Write each power of i as i, - 1, - i, or 1.
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Verified step by step guidance

1. Step 1: Recall the cyclical nature of powers of i. The powers of i repeat in a cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats for higher powers.
2. Step 2: To determine the value of i^114, divide the exponent (114) by 4 and find the remainder. This is because the cycle repeats every 4 powers.
3. Step 3: Perform the division 114 ÷ 4. The quotient is 28, and the remainder is 2. This means i^114 is equivalent to i^2.
4. Step 4: Refer back to the cycle of powers of i. From the cycle, i^2 = -1.
5. Step 5: Conclude that i^114 simplifies to -1 based on the cyclical pattern of powers of i.

Key Concepts

Powers of i

The imaginary unit i is defined as the square root of -1. Its powers cycle through four distinct values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cyclical pattern repeats every four powers, which is crucial for simplifying higher powers of i.

Modulus and Division

To simplify powers of i, we can use the modulus of the exponent. Since the powers of i repeat every four terms, we can find the equivalent power by calculating the exponent modulo 4. For example, i^114 can be simplified by finding 114 mod 4, which helps determine the corresponding value in the cycle.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. Understanding complex numbers is essential for working with i, as it allows for the manipulation and interpretation of expressions involving imaginary units in various mathematical contexts.