Simplify each power of i. i^11

Verified step by step guidance

1. insert step 1> Identify the pattern of powers of i: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1.
2. insert step 2> Notice that the powers of i repeat every four terms: i^5 = i, i^6 = -1, i^7 = -i, i^8 = 1, and so on.
3. insert step 3> To simplify i^{11}, divide 11 by 4 to find the remainder: 11 ÷ 4 = 2 remainder 3.
4. insert step 4> Use the remainder to determine the equivalent power of i: Since the remainder is 3, i^{11} is equivalent to i^3.
5. insert step 5> Refer to the pattern: i^3 = -i, so i^{11} simplifies to -i.

Key Concepts

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0. Understanding 'i' is crucial for simplifying powers of 'i' and working with complex numbers.

Powers of i

The powers of 'i' follow a cyclical pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats every four powers, meaning that any power of 'i' can be simplified by finding the remainder when the exponent is divided by 4. This property is essential for simplifying higher powers of 'i', such as i^11.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus. In the context of simplifying powers of 'i', we use modulus 4 to determine the equivalent lower power of 'i'. For example, to simplify i^11, we calculate 11 mod 4, which equals 3, leading to the simplification i^11 = i^3 = -i.