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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblema 98
Capitolo 2, Problema 98

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

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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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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.
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Dividing 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 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.
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Powers of i

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.
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Multiplying Complex Numbers
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