In Exercises 95–99, perform the indicated operations and write the result in standard form. (i⁸⁵ - i⁸³)/i⁴⁵

검증된 단계별 안내

Step 1: Recall the powers of the imaginary unit i. The powers of i repeat in a cycle: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats every four powers. Use this property to simplify i^85, i^83, and i^45.

Step 2: Simplify i^85 by dividing 85 by 4 and finding the remainder. The remainder determines the equivalent power of i within the cycle. Similarly, simplify i^83 and i^45 using the same method.

Step 3: Substitute the simplified values of i^85, i^83, and i^45 into the expression (i^85 - i^83)/i^45.

Step 4: Perform the subtraction in the numerator and simplify the result. Then divide the simplified numerator by the simplified denominator.

Step 5: Write the final result in standard form, which is a + bi, where a and b are real numbers.

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주요 개념

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Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically expressed in the form a + bi, where 'i' is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for manipulating expressions involving 'i' and performing operations such as addition, subtraction, multiplication, and division.

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 cyclical nature means that any power of 'i' can be simplified by reducing the exponent modulo 4. Recognizing this pattern is crucial for simplifying expressions involving higher powers of 'i' in the given problem.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' is the real part and 'b' is the imaginary part. When performing operations with complex numbers, it is important to express the final result in this form for clarity and consistency. This involves combining like terms and ensuring that the imaginary unit 'i' is properly accounted for in the final expression.

In Exercises 95–99, perform the indicated operations and write the result in standard form. (i85 - i83)/i45