Problem 96 Simplify. $$ 5 i^{5}+4 i^{3}... [FREE SOLUTION]

Chapter 10: Problem 96

Simplify. $$ 5 i^{5}+4 i^{3} $$

Short Answer

Expert verified

The expression simplifies to $i$.

Step by step solution

- Review properties of imaginary unit

Recall that the imaginary unit, denoted by $i$, satisfies the equation $i^2 = -1$. Therefore, higher powers of $i$ can be expressed in terms of $i$, $-1$, $-i$, or $1$.

- Simplify $i^5$

Notice that $i^5 = i^{4+1} = (i^4) \times i$. Since $i^4 = (i^2)^2 = (-1)^2 = 1$, we have $i^5 = 1 \times i = i$.

- Simplify $i^3$

Similarly, $i^3 = i^{2+1} = (i^2) \times i = (-1) \times i = -i$.

- Substitute and combine terms

Substitute the results into the original expression: \[ 5i^5 + 4i^3 \] becomes \[ 5i + 4(-i) \]. Simplify by combining like terms: \[ 5i - 4i = i \].

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

properties of imaginary unit

Let's start by understanding the imaginary unit, denoted as $i$. The imaginary unit is a fundamental concept in complex numbers and has the key property that $i^2 = -1$.
This special property allows us to rewrite higher powers of $i$ based on a repeating pattern. Here are the main properties to keep in mind:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

Notice that after every four powers, the values repeat. This will help us greatly when simplifying expressions involving powers of $i$.
Understanding these properties simplifies calculations and makes it easier to work with complex numbers.

higher powers of i

Now, let's apply the properties of $i$ to simplify higher powers of $i$. Given that $i^4 = 1$, any power of $i$ can be expressed in terms of these basic four values. For example, in the exercise, we encountered $i^5$ and $i^3$.
Here鈥檚 how to simplify them:

$i^5 = i^{4+1} = (i^4) \times i = 1 \times i = i$
$i^3 = i^{2+1} = (i^2) \times i = -1 \times i = -i$

By breaking down the exponents and using the pattern, we can simplify these powers easily. This method ensures that we never have to deal with exponents higher than 4, making our calculations straightforward and more manageable.

combining like terms

Finally, let's see how to combine like terms in complex number expressions. In algebra, like terms have the same variables raised to the same power and can be added or subtracted directly. This concept also applies to expressions involving $i$.
Consider the example from the exercise:

Initially, the expression was $5i^5 + 4i^3$.
We simplified it to $5i - 4i$.

Now, we combine the like terms, both of which involve $i$:

$5i - 4i = 1i = i$.

Therefore, the expression simplifies to $i$. Remember, combining like terms helps reduce complex expressions to simpler forms which are easier to interpret and use.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Simplify. $$ 5 i^{5}+4 i^{3} $$

Short Answer

Step by step solution

- Review properties of imaginary unit

- Simplify \(i^5\)

- Simplify \(i^3\)

- Substitute and combine terms

Key Concepts

properties of imaginary unit

higher powers of i

combining like terms

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Probability and Statistics

Pure Maths

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.