In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i^44

Verified step by step guidance

1. <insert step 1> Start by understanding the powers of the imaginary unit i. The powers of i cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1.
2. <insert step 2> Notice that i^4 = 1, which means every fourth power of i is 1. Therefore, any power of i can be reduced by dividing the exponent by 4 and using the remainder to determine the equivalent power of i.
3. <insert step 3> Divide the exponent 44 by 4 to find the remainder: 44 ÷ 4 = 11 with a remainder of 0.
4. <insert step 4> Since the remainder is 0, i^44 is equivalent to i^0, which is 1, because any number to the power of 0 is 1.
5. <insert step 5> Therefore, i^44 simplifies to 1.>

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 with respect to 4. For example, to find i^44, we calculate 44 mod 4, which equals 0. This indicates that i^44 corresponds to i^0, which is equal to 1, following the established cycle of powers.

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 powers of i, as they form the basis for operations involving imaginary units in algebra.