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

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 96

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

Short Answer

Expert verified

The answer is $i$.

Step by step solution

Recall the powers of imaginary unit

Understand that the imaginary unit, denoted as $i$, has cyclical powers. The key powers to remember are: $i^1 = i$$i^2 = -1$$i^3 = -i$$i^4 = 1$Notice that these powers repeat every four exponents: $i^5 = i$, $i^6 = -1$, and so on.

Apply the cyclical pattern to simplify the terms

Rewrite each term using the cyclical pattern. For the term $5i^5$, since $i^5 = i$, it simplifies to $5i$. For $4i^3$, since $i^3 = -i$, it simplifies to $4(-i) = -4i$.

Combine like terms

Combine the simplified terms: $5i - 4i$. Since both terms have the same imaginary unit, add the coefficients: $5 - 4 = 1$. Thus, the simplified form is $i$.

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Key Concepts

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

powers of imaginary unit

In this exercise, understanding the powers of the imaginary unit, denoted as $i$, is crucial. The imaginary unit $i$ satisfies the equation $i^2 = -1$. From this, we can derive more powers of $i$:

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

The pattern of powers repeats every four exponents. For instance, $i^5$ cycles back to $i$. This cyclical nature helps in simplifying higher powers of $i$ without direct computation.

cyclical pattern of imaginary unit

The cyclical pattern is a key method to reduce higher powers of $i$ down to simpler forms. The pattern is:

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

After $i^4$, the cycle repeats. For example, $i^5 = i^{(4+1)} = i$, because every fourth power brings us back to 1 (the start of the cycle), and adding 1 gives $i$. Similarly, $i^6 = -1$, because we effectively have $i^2$ again. Using this cyclical pattern makes it easy to simplify expressions involving higher powers of $i$.

combining like terms

Once the terms in the expression involving imaginary units are simplified, the next step is to combine like terms. For example, in the expression $5i^5 + 4i^3$:

First, we reduce $i^5$ to $i$ using the cyclical pattern, simplifying to $5i$.
Next, we reduce $i^3$ to $-i$, simplifying to $4(-i)$ or $-4i$.

Now, we have $5i - 4i$. Here, both terms are like terms because they have the same imaginary unit ($i$). Combining them means adding their coefficients: $5 - 4 = 1$. Therefore, the simplified form of the original expression is $i$.

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91影视

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

Short Answer

Step by step solution

Recall the powers of imaginary unit

Apply the cyclical pattern to simplify the terms

Combine like terms

Key Concepts

powers of imaginary unit

cyclical pattern of imaginary unit

combining like terms

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