Problem 39 Find the indicated powers of com... [FREE SOLUTION]

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Chapter 7: Problem 39

Find the indicated powers of complex numbers. $$i^{9}$$

Short Answer

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Step by step solution

Define the imaginary unit

The imaginary unit, denoted as $i$, is defined as $i = \sqrt{-1}$. This means $i^2 = -1$.

Understand the repeating cycle 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 continues for every four exponents.

Find the equivalent power in the cycle

To find $i^9$, recognize the repeating cycle. Divide 9 by 4. The remainder will give the relevant power in the cycle: $9 \text{ mod } 4 = 1$. This means \i^9\ is equivalent to \i^1\.

Determine the value from the cycle

Since $i^9$ is equivalent to $i^1$, and we know from the cycle that $i^1 = i$, thus $i^9 = i$.

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Key Concepts

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

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers. It is denoted by the symbol $ i $. The most important property of the imaginary unit is that it is defined as the square root of negative one. In mathematical terms, this is written as: $ i = \sqrt{-1} $. This definition leads us to the equation $ i^2 = -1 $. This property of $ i $ is crucial for understanding complex numbers and their behavior.

Powers of i

Understanding the powers of the imaginary unit $ i $ is essential for solving many complex number problems. The powers of $ i $ follow a simple cyclical pattern. As you increase the exponent, you'll find that the results repeat every four exponents. Here鈥檚 the cycle:

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

After the 4th power, the cycle repeats. So, for example, $ i^5 $ would be the same as $ i^1 $, $ i^6 $ the same as $ i^2 $, and so on. To find higher powers, you can use the modulus operation, which we'll discuss next.

Modulus Operation

The modulus operation helps us simplify higher powers of $ i $ by taking advantage of the repeating cycle. Modulus is a mathematical operation that finds the remainder after division of one number by another. In the context of complex numbers, we use the modulus operation to reduce higher powers of $ i $ to a smaller, manageable number within the cycle. For example, to find $ i^9 $, you divide 9 by 4 (since the cycle repeats every 4):

$ 9 \text{ mod } 4 = 1 $ because 9 divided by 4 is 2 with a remainder of 1.

Therefore, $ i^9 $ is equivalent to $ i^1 $, which is simply $ i $. This technique allows you to handle even large exponents with ease, by leveraging the cyclical nature of the powers of $ i $.

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91影视

Find the indicated powers of complex numbers. $$i^{9}$$

Short Answer

Step by step solution

Define the imaginary unit

Understand the repeating cycle of powers of \(i\)

Find the equivalent power in the cycle

Determine the value from the cycle

Key Concepts

Imaginary Unit

Powers of i

Modulus Operation

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