Skip to main content
Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 98
Chapter 2, Problem 98

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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
04:22
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.
Recommended video:
04:10
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.
Recommended video:
05:02
Multiplying Complex Numbers
Related Practice
Textbook Question

Solve each equation in Exercises 96–102 by the method of your choice. 2√(x-1) = x

Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice. x2−6x+13=0x^2 - 6x + 13 = 0

5
views
Textbook Question

In Exercises 99–106, solve each equation. [(3 + 6)2 ÷ 3] × 4 = - 54 x

Textbook Question

In Exercises 91–100, find all values of x satisfying the given conditions. y=(x−5)32andy=125y = (x - 5)^{\(\frac{3}{2}\)} \(\quad\) \(\text{and}\) \(\quad\) y = 125

Textbook Question

Evaluate x2 - (xy - y) for x satisfying 3(x + 3)/5 = 2x + 6 and y satisfying - 2y - 10 = 5y + 18.

1
views
Textbook Question

Use interval notation to represent all values of x satisfying the given conditions. y = 2x - 11 + 3(x + 2) and y is at most 0