Problem 89 Simplify. $$i^{9}$$... [FREE SOLUTION]

91影视

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 89

Simplify. $$i^{9}$$

Short Answer

Expert verified

Step by step solution

- Understand the Powers of i

The imaginary unit, denoted as $i$, follows a cyclical pattern in its powers. The key pattern to remember is: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$. This cycle repeats every four powers.

- Determine the Cycle Position

Find where the exponent 9 falls within the cycle by dividing 9 by 4 and finding the remainder. The remainder will indicate the position in the cycle. $ 9 \,\text{divided by}\, 4 = 2 \,\text{remainder}\, 1 $.

- Identify the Equivalent Term

Since the remainder is 1, $i^9$ is equivalent to $i^1$.

- Write the Simplified Form

From the power pattern, $i^1 = i$. Therefore, $i^9 = i$.

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Key Concepts

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

Cyclical Pattern of i

Powers of the imaginary unit, denoted as $i$, follow a consistent and repeating pattern. This pattern is crucial for simplifying expressions with powers of $i$. The cyclical behavior is as follows:

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

Once you reach $i^4$, the cycle repeats. This means $i^5 = i$, $i^6 = -1$, and so forth. Recognizing this loop helps streamline the process of simplifying higher powers of $i$.
Whenever you encounter any power of $i$, you can simplify it by determining its position within this consistent 4-step cycle.

Simplifying Powers of i

To simplify powers of $i$, you use the cyclical pattern you just learned. Let's consider the exercise example, $i^9$. Knowing the cyclical pattern, you can determine the equivalent position within the cycle:

First, notice that $i^9$ is beyond $i^4$, so it repeats the initial cycle.
Next, find how many complete cycles of 4 fit into 9 and identify the remainder. This will tell you which power within the first cycle $i^9$ correlates to.

By following this approach:

Divide 9 by 4, which gives you a quotient of 2 and a remainder of 1.
Therefore, $i^9$ corresponds to $i^1$ because the remainder is 1.

Recognize that finding these remainders is pivotal for simplification. Thus, $i^9$ simplifies to $i$.

Remainder in Division

The remainder is essential when simplifying powers of $i$ because it indicates the position within the 4-step cycle. Here鈥檚 how it works:

When you have a power of $i$, divide the exponent by 4.
The remainder of this division tells you which term in the cycle you are dealing with.

For the exercise given, divide 9 by 4:

9 divided by 4 gives 2 with a remainder of 1.

Therefore, $i^9$ simplifies to $i^1$. This method is efficient and can be applied universally to any power of $i$. Remember:

If the remainder is 0, the power corresponds to $i^4$, which equals 1.
If the remainder is 1, it corresponds to $i^1$, which equals $i$.
If the remainder is 2, it corresponds to $i^2$, which equals -1.
If the remainder is 3, it corresponds to $i^3$, which equals -i.

Mastering this concept helps tremendously in simplifying any powers of $i$.

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91影视

Simplify. $$i^{9}$$

Short Answer

Step by step solution

- Understand the Powers of i

- Determine the Cycle Position

- Identify the Equivalent Term

- Write the Simplified Form

Key Concepts

Cyclical Pattern of i

Simplifying Powers of i

Remainder in Division

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