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Chapter 2: Problem 88

In Exercises 79 - 88, simplify the complex number and write it in standard form. \( (-i)^6 \)

Short Answer

Expert verified
The simplification of expression \(-i^6\) gives \(1\).

Step by step solution

01

Acknowledge the properties of \(i\)

The basis of the imaginary unit \(i\) is such that \(i = \sqrt{-1}\). Based on this, we get further properties: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\) and \(i^4 = 1\). Note that the powers of \(i\) subsequently repeat every four powers.
02

Find the result of \(i^6\)

\(i^6\) is the same as \(i^{4 \times 1} \times i^2\), as the powers add up to 6. \(i^4\) equals to 1 (since the powers of \(i\) repeats every four powers), and \(i^2\) equals to -1. Therefore, \(i^6 = 1 \times -1 = -1\).
03

Evaluate \(-i^6\)

We have determined that \(i^6 = -1\), so \(-i^6 = -(-1) = 1\).

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Key Concepts

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

Imaginary Unit
Imagine you are exploring a new type of number that goes beyond the familiar landscape of whole numbers, fractions, and irrationals - welcome to the realm of the imaginary unit, denoted by the symbol \(i\). This enigmatic entity is defined as the square root of negative one: \(i = \sqrt{-1}\).

Now, why is this significant? In the standard number system, the square root of a negative number doesn't have a place, but in complex analysis, \(i\) bridges this gap, providing us with a robust extension of the real numbers. The concept of the imaginary unit allows us to perform calculations and solve equations that would otherwise be unsolvable within the realm of real numbers alone.
Standard Form of Complex Numbers
Complex numbers are a blend of real and imaginary numbers, usually expressed in 'standard form' which looks like this: \(a + bi\), where \(a\) and \(b\) are real numbers. In this duo, \(a\) is the 'real part', while \(bi\) is the 'imaginary part'.

For instance, in the complex number \(3 + 4i\), the number 3 is our real part and \(4i\) is the imaginary part. This form allows us to neatly organize and handle complex numbers for a variety of operations, including addition, subtraction, multiplication, and even division, by providing a consistent structure for representation.
Properties of i
The imaginary unit \(i\) has fascinating cyclical properties keyed to its powers. To begin, as we know, \(i = \sqrt{-1}\). From here, squaring both sides yields \(i^2 = -1\). Taking it a step further, \(i^3 = i^2 \times i = -i\), and interestingly, \(i^4 = (i^2)^2 = 1\).

What's compelling is that after the fourth power, these values begin to repeat. So \(i^5\) would revert to \(i\), \(i^6\) to \(-1\), and so on. This cyclical pattern proves highly useful in simplifying expressions with higher powers of \(i\), as showcased in our exercise.
Simplifying Complex Numbers
Simplifying higher powers of \(i\) calls for a methodical approach which makes use of the properties discussed earlier. Given a power like \(i^6\), we divide that exponent by 4 and observe the remainder, or alternatively think of it as cycles of four. Since \(6\) is greater than \(4\), we break down \(i^6\) as \(i^{4+2}\), which via properties, simplifies to \(i^4 \times i^2 = 1 \times -1 = -1\).

Understanding this repetition allows us to tackle even more complex expressions and maintain our operations within the realm of manageable calculations. Through practice and application of these properties, simplifying complex numbers becomes a clear and logical process.

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Most popular questions from this chapter

In Exercises 83 - 86, (a) find the interval(s) for \( b \) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. \( 2x^2 + bx + 5 = 0 \)

In Exercises 83 - 86, (a) find the interval(s) for \( b \) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. \( 3x^2 + bx + 10 = 0 \)

In Exercises 69 - 72, use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. \( g(x) = \dfrac{1 + 3x^2 - x^3}{x^2} \)

Is every rational function a polynomial function? Is every polynomial function a rational function? Explain.

In Exercises 37 - 40, use a graphing utility to graph the equation. Use the graph to approximate the values of that satisfy each inequality. Equation \( y = \dfrac{1}{2}x^2 - 2x + 1 \) Inequalities (a) \( y \le 0 \) (b) \( y \ge 7 \)
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