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Chapter 3: Problem 82

Simplify. $$i^{24}$$

Short Answer

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Step by step solution

01

Understand the Concept of Imaginary Unit Power Cycles

The imaginary unit, denoted as \(i\), has a cyclical pattern when raised to successive powers. Specifically, \(i\) cycles every four powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\). This pattern repeats with every four powers.
02

Recognize the Cyclical Pattern of i

From the cyclical pattern, we know that every fourth power of \(i\) is equal to 1. Therefore, \(i^4 = 1\), \(i^8 = (i^4)^2 = 1^2 = 1\), and so on.
03

Simplify the Given Power using the Cyclical Pattern

To simplify \(i^{24}\), recognize that 24 is a multiple of 4. Thus, \(i^{24} = (i^4)^6\). Given that \(i^4 = 1\), it follows that \(i^{24} = 1^6 = 1\).

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Key Concepts

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

cyclical pattern of i
The imaginary unit, represented by the letter \(i\), follows a cyclical pattern when raised to powers. This cycle repeats every four powers, making it easier to simplify larger exponents of \(i\). Here鈥檚 the pattern:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)

Once we reach \(i^4\), the cycle repeats: \(i^5 = i\), \(i^6 = -1\), and so forth. Understanding this cyclical behavior is crucial because it allows us to simplify any power of \(i\). It鈥檚 a nice little shortcut that saves a lot of time!
simplifying powers of i
To simplify powers of \(i\), you should use the cyclical pattern. Here鈥檚 how:
Identify where your power lies within the cycle of four. For example, to simplify \(i^{24}\), recognize that 24 is a multiple of 4. If the exponent is not a multiple of 4, divide the exponent by 4 to find the remainder.
For instance, for \(i^{10}\):
  • Divide 10 by 4
  • The remainder is 2
  • This tells us \(i^{10} = i^2\)
Since \(i^2 = -1\), we have \(i^{10} = -1\). This method works for any exponent, simplifying the task of working with \(i\).
multiple of 4 in powers
Understanding that the power of \(i\) cycles every four, simplifies the process significantly, especially when the exponent is a multiple of 4. Consider an example like \(i^{24}\). Since 24 is a multiple of 4, we can write it as \(24 = 4 \times 6\). This means:
\(i^{24} = (i^4)^6 = 1^6 = 1\)
Therefore, any exponent that is a multiple of 4 will cycle back to 1. This trick helps bypass lengthy calculations and gives a direct answer. Applying this knowledge can make working with imaginary units straightforward and less intimidating.

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Most popular questions from this chapter

Fill in the blank with the correct term. Some of the given choices will not be used. distance formula, midpoint formula, function, relation, \(x\) -intercept, y-intercept, perpendicular, parallel ,horizontal lines, vertical lines,symmetric with respect to the \(x\) -axis, symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, increasing, decreasing, constant _____ are given by equations of the type \(y=b,\) or \(f(x)=b\).

In business, profit is the difference between revenue and cost; that is, $$ \begin{array}{l} \text { Total profit }=\text { Total revenue }-\text { Total cost, } \\ P(x)=R(x)-C(x) \end{array} $$ where \(x\) is the number of units sold. Find the maximum profit and the number of units that must be sold in order to yield the maximum profit for each of the following. $$R(x)=50 x-0.5 x^{2}, C(x)=10 x+3$$

Find the zero of the function. $$f(x)=-3 x+9$$

a) Find the vertex. b) Determine whether there is a maximum or a minimum value and find that value. c) Find the range. d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing. $$f(x)=-3 x^{2}+24 x-49$$

A model rocket is launched with an initial velocity of \(150 \mathrm{ft} / \mathrm{sec}\) from a height, of \(40 \mathrm{ft}\). The function \(s(t)=-16 t^{2}+150 t+40\) gives the height of the rocket, in feet, \(t\) seconds after it has been launched. Determine the time at which the rocket reaches its maximum height and find the maximum height.
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