Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

Previous problemNext problem

2:06 minutes

Problem 36

Textbook Question

Simplify each power of i. 1/i^17

Verified step by step guidance

Recall that the imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), and it has a cyclical pattern: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then it repeats.

To simplify \( i^{17} \), first find the remainder when 17 is divided by 4, since the powers of \( i \) repeat every 4.

Calculate \( 17 \div 4 \) which gives a quotient of 4 and a remainder of 1.

Since the remainder is 1, \( i^{17} = i^1 = i \).

Now, simplify \( \frac{1}{i} \) by multiplying the numerator and the denominator by \( i \) to rationalize the denominator: \( \frac{1}{i} \times \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

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 working with powers of 'i', as they exhibit unique properties that differ from real 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 cycle repeats every four powers, which means any power of 'i' can be simplified by reducing the exponent modulo 4. Recognizing this pattern is crucial for simplifying expressions involving 'i'.

Reciprocal of a Complex Number

The reciprocal of a complex number, such as 1/i^n, can be simplified by multiplying the numerator and denominator by the conjugate of the denominator. This process helps eliminate the imaginary unit from the denominator, making it easier to work with complex fractions. Understanding how to manipulate complex numbers is key to simplifying expressions like 1/i^17.