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 57a
Chapter 2, Problem 57a

Write each power of i as i, - 1, - i, or 1.
i114

Verified step by step guidance
1
Step 1: Recall the cyclical nature of powers of i. The powers of i repeat in a cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats for higher powers.
Step 2: To determine the value of i^114, divide the exponent (114) by 4 and find the remainder. This is because the cycle repeats every 4 powers.
Step 3: Perform the division 114 ÷ 4. The quotient is 28, and the remainder is 2. This means i^114 is equivalent to i^2.
Step 4: Refer back to the cycle of powers of i. From the cycle, i^2 = -1.
Step 5: Conclude that i^114 simplifies to -1 based on the cyclical pattern of powers of i.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

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.
Recommended video:
04:10
Powers of i

Modulus and Division

To simplify powers of i, we can use the modulus of the exponent. Since the powers of i repeat every four terms, we can find the equivalent power by calculating the exponent modulo 4. For example, i^114 can be simplified by finding 114 mod 4, which helps determine the corresponding value in the cycle.
Recommended video:
05:10
Higher Powers of i

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 i, as it allows for the manipulation and interpretation of expressions involving imaginary units in various mathematical contexts.
Recommended video:
04:22
Dividing Complex Numbers
Related Practice
Textbook Question

Solve each equation in Exercises 58–59 by factoring. 2x^2 +15x = 8

Textbook Question

Write each power of i as i, - 1, - i, or 1.

i44

Textbook Question

In Exercises 59–94, solve each absolute value inequality. |x| < 3

Textbook Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. (y−8y)2+5(y−8y)−14=0\(\left\)( y - \(\frac{8}{y}\) \(\right\))^2 + 5 \(\left\)( y - \(\frac{8}{y}\) \(\right\)) - 14 = 0

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. x2+3x−1=0x^2 + 3x - 1 = 0

3
views
Textbook Question

In Exercises 51–58, solve each compound inequality. - 3 ≤ (2/3)x - 5 < - 1