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Chapter 10: Problem 87

In Exercises \(85-100,\) simplify each expression. $$i^{11}$$

Short Answer

Expert verified
\(-i\)

Step by step solution

01

Identify the Pattern of Powers of i

The powers of the imaginary unit \(i\) follow a cyclic pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), and then the pattern repeats.
02

Find the Remainder of the Power Divided by 4

The number 11 (the power in \(i^{11}\)) divided by 4 gives a quotient of 2 and a remainder of 3. Therefore, \(i^{11}\) has the same value as \(i^3\) in the cyclic pattern.
03

Apply the Identified Pattern

According to the pattern we identified in Step 1, \(i^3 = -i\). Hence, \(i^{11}\) simplifies to \(-i\).

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Key Concepts

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

Imaginary Unit
At the heart of simplifying complex numbers lies the concept of the imaginary unit, commonly represented by the symbol i. When we learn about numbers and algebra, we are initially introduced to real numbers which can be plotted on a simple number line. However, the square root of a negative number cannot be found on this line, as no real number multiplied by itself gives a negative result. The imaginary unit i is defined as i = \(\text{sqrt{-1}}\). Thus, it represents what we would otherwise consider impossible in the realm of real numbers.

Cyclic Pattern of Powers of i
Understanding how to handle the powers of the imaginary unit requires recognizing the cyclic pattern they follow. When taking powers of i, you'll notice that it cycles through four different outcomes. This arises from the definition of i and proceeds as follows:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
After reaching \(i^4\), the pattern repeats indefinitely. This means \(i^5\) is the same as \(i^1\), \(i^6\) is the same as \(i^2\), and so on. A key takeaway from this cyclic nature is that for any integer power of i, say \(i^n\), we can find an equivalent power within the first four powers by merely looking at the remainder when n is divided by 4.

Simplify Powers of i
To simplify powers of i, such as \(i^{11}\), we use the cyclic pattern of powers of i. We divide the exponent by 4 and focus on the remainder, because through the pattern, we know that every power of i eventually boils down to one of the first four powers of i. In the case of \(i^{11}\), when we divide 11 by 4, we're left with a remainder of 3. Therefore, \(i^{11}\) simplifies to the same value as \(i^3\), which, based on our pattern, we've previously identified as \(-i\). This insightful method allows us to deal with even the most intimidating powers of i with ease, showcasing the beauty and simplicity in complex numbers.

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Most popular questions from this chapter

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{17}{\sqrt{10}-2}$$

Use a graphing utility to solve each radical equation. Graph each side of the equation in the given viewing rectangle. The equation's solution set is given by the \(x\) -coordinate(s) of the point (s) of intersection. Check by substitution. $$\begin{aligned} &\sqrt{2 x+2}=\sqrt{3 x-5}\\\ &[-1,10,1] \text { by }|-1,5,1| \end{aligned}$$

In Exercises \(129-132\), determine if each operation is performed correctly by graphing the function on each side of the equation with your graphing utility. Use the given viewing rectangle. The graphs should be the same. If they are not, correct the right side of the equation and then use your graphing utility to verify the correction. $$\begin{aligned} &(\sqrt{x}-1)(\sqrt{x}-1)=x+1\\\ &[0,5,1] \text { by }[-1,2,1] \end{aligned}$$

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{35}{5 \sqrt{2}-3 \sqrt{5}}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. After squaring both sides of a radical equation, the only solution that I obtained was extraneous, so \(\varnothing\) must be the solution set of the original equation.
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