In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i³¹

검증된 단계별 안내

Step 1: Recall the cyclical nature of powers of i. The powers of i repeat in a cycle of four: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats for higher powers of i.

Step 2: To determine the value of i^31, divide the exponent (31) by 4 and find the remainder. This is because the cycle repeats every 4 powers.

Step 3: Perform the division: 31 ÷ 4. The quotient is 7, and the remainder is 3. This means i^31 is equivalent to i^3.

Step 4: Refer back to the cycle of powers of i. From the cycle, i^3 = -i.

Step 5: Conclude that i^31 simplifies to -i based on the cyclical pattern of powers of i.

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주요 개념

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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^31, we calculate 31 mod 4, which gives us a remainder of 3. This means i^31 is equivalent to i^3, allowing us to simplify the expression effectively.

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 many operations in algebra involving imaginary units.

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i31