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Chapter 2: Problem 88

Simplify the complex number and write it in standard form. \((-i)^{6}\)

Short Answer

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Step by step solution

01

Calculate Power

Since the base is negative \(i\) and \(i^2 = -1\), then we follow the following pattern for \(i^n\): \[ i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, i^6 = -1 \ldots\] The pattern repeats every four steps. So, \((-i)^6\) is the same as \(i^6\), because \(-1^{6}\) is 1. Therefore, \(i^6 = i^{6 \mod 4} = i^2 = -1 \].
02

Write the Result in Standard Form

The answer is already in standard form for complex numbers, which is 'a+bi', where a is the real part, and b is the imaginary part. Here, the real part is -1, and the imaginary part is 0. Thus, the standard form is -1.

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Key Concepts

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

Understanding Standard Form
When dealing with complex numbers, it is crucial to write them in the standard form, which is expressed as \(a + bi\). Here, \(a\) is the real part, and \(b\) is the imaginary part. This format helps simplify complex arithmetic and makes it easier to identify and make comparisons. For example, in the exercise provided, the original expression is transformed into \(-1\), which means there is no imaginary component. In standard form, this could be written as \(-1 + 0i\). This adjusts the solution into a recognizable structure where both the real and imaginary parts are clear.
Diving into the Powers of i
To simplify expressions involving powers of \(i\), we should understand the rotation pattern of \(i\) when multiplied by itself. Expressed in simple cycles, these powers help solve many problems that involve complex numbers. The key transformations are:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
This cycle repeats itself every four powers. Therefore, knowing this repeating sequence allows you to simplify expressions quickly. For instance, for \((-i)^6\), rewriting this as \(i^6\) after eliminating the negative sign yields \(i^6 = i^{6 \mod 4} = i^2\). Hence, this simplifies directly to \(-1\). Breaking down complex powers in this manner makes them much more manageable.
Insights into the Imaginary Unit
The imaginary unit \(i\) is foundational in complex number theory. It is defined by its property \(i^2 = -1\). This definition is what makes it distinct and enables us to extend the number system beyond the real numbers. Imaginary units come into play whenever square roots of negative numbers are involved, offering a solution where none previously existed in real numbers.
  • The term 'imaginary' might be misleading鈥攊t doesn't make such numbers any less real or important in mathematics or engineering.
  • They allow the representation and solution of equations that would otherwise have no solution in the realm of real numbers, greatly expanding mathematical modeling capabilities.
This helps in solving quadratic equations, performing transformations in physics, and is used extensively in electrical engineering for analyzing circuits.

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Most popular questions from this chapter

Solve the inequality and graph the solution on the real number line. \(\frac{5}{x-6}>\frac{3}{x+2}\)

Solve the inequality and graph the solution on the real number line. \(\frac{3 x-5}{x-5} \geq 0\)

Use a graphing utility to graph the equation. Use the graph to approximate the values of \(x\) that satisfy each inequality. \(y=\frac{5 x}{x^{2}+4} \quad\) (a) \(y \geq 1 \quad\) (b) \(y \leq 0\)

Solve the inequality. (Round your answers to two decimal places.) \(0.4 x^{2}+5.26<10.2\) 70\. \(\frac{2}{3.1 x-3.7}>5.8\)

Solve the inequality and graph the solution on the real number line. \(2 x^{3}+13 x^{2}-8 x-46 \geq 6\)
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