Perform the indicated operations and write the result in standard form. (i⁹⁸ - i⁹⁴)/i⁴⁹

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 4 powers. Use this property to simplify the powers of i in the numerator and denominator.

Step 2: Simplify i^98. Since the powers of i repeat every 4, divide 98 by 4 and find the remainder. The remainder determines the equivalent power of i. Similarly, simplify i^94 using the same process.

Step 3: Simplify the denominator i^49. Again, divide 49 by 4 and find the remainder to determine the equivalent power of i.

Step 4: Substitute the simplified values of i^98, i^94, and i^49 into the expression (i^98 - i^94)/i^49.

Step 5: Perform the subtraction in the numerator and simplify the division by the denominator. Write the final result in standard form (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, 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 cycle repeats every four powers, which allows for simplification of higher powers of 'i' by reducing them modulo 4. Recognizing this pattern is crucial for simplifying expressions involving powers of 'i' in the given problem.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary part. When performing operations with complex numbers, it is important to express the final result in this standard form to clearly identify the real and imaginary components, facilitating further analysis or interpretation.

Perform the indicated operations and write the result in standard form. (i98 - i94)/i49