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Chapter 10: Problem 95

In Exercises \(85-100,\) simplify each expression. $$(-i)^{4}$$

Short Answer

Expert verified
The simplification of \( (-i)^{4} \) is 1.

Step by step solution

01

Substitute the Value of \(i^2\)

The first part is to understand the nature of the imaginary unit \(i\). Remember that \(i^2 = -1\). Knowing this, we can raise \(i\) to the power of 4 by multiplying two instances of \(i^2\) together. It becomes \( (-i)^4 = ((-i)^2)^2 = ((-1)^2 * i^2)^2 \).
02

Simplify the Expression

The second part is to simplify the expression. \( ((-1)^2 * i^2)^2 = (1* -1 )^2 =-1^2 = 1 \).
03

Final Result

So, the simplification of the given expression \( {-i}^4 \) is 1.

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Key Concepts

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

Imaginary Unit
Complex numbers are a fascinating realm of mathematics, and at the heart of complex numbers lies the imaginary unit, denoted by the symbol \( i \). Essentially, \( i \) is defined such that \( i^2 = -1 \). This concept may initially seem puzzling because we aren't used to taking the square root of negative numbers.

Here's where the imaginary unit comes into play:
  • The imaginary unit \( i \) allows us to work with numbers that include roots of negative numbers, transforming these roots into manageable elements in mathematical expressions.
  • When you encounter \( i \) in an expression, it signifies the inclusion of an imaginary component, widening the scope of possible solutions in equations.
The imaginary unit isn't just a strange mathematical object; it adds a whole new dimension to numbers, quite literally when representing complex numbers as a plane. By embracing \( i \), mathematicians can solve problems that are otherwise impossible with just real numbers.
Exponents
Understanding exponents is vital in simplifying expressions, especially when involving complex numbers like \( (-i)^4 \). Exponents denote multiplying a number by itself a certain number of times. For example, \( a^n \) means multiplying \( a \) by itself \( n \) times.

In complex numbers and expressions:
  • Raising \( i \) to various powers reveals a cycle, due to its definition \( i^2 = -1 \).
  • Consider \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Notice how every fourth power of \( i \) loops back to 1.
In our specific example, \( (-i)^4 \), observe how this knowledge simplifies our process:
  • We recognize that \( (-i)^2 = (-1)^2 \times i^2 = 1 \times (-1) = -1 \).
  • Therefore, \( (-i)^4 = ((-i)^2)^2 = (-1)^2 = 1 \).
Mastering how to work with exponents in the context of the imaginary unit empowers you to uncover elegant solutions in complex arithmetic.
Simplification of Expressions
Simplifying expressions, especially those with complex numbers, requires applying rules judiciously to reduce a complex appearing concept into a more understandable form. The simplification process is systematic:

  • Identify known rules and properties: Like how you recognize \( i^2 = -1 \) as a fundamental property of \( i \).
  • Apply logical steps: Use algebraic manipulations to break down expressions into simpler components, ensuring each step logically follows the previous one.
  • Keep track of signs: Complex numbers involve careful attention to negative and positive signs as these influence the final outcome significantly.
For our case \( (-i)^4 \):
  • Start by recognizing \( (-i)^2 = (-1)^2 \times i^2 = -1 \), which simplifies to use in subsequent steps.
  • Raise the result to the power required, here \( ((-i)^2)^2 = (-1)^2 = 1 \).
The aim of simplification is not just to find a solution, but to express mathematical relationships in their simplest and most beautiful forms. By mastering these processes, any complex expression becomes approachable and solvable.

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Most popular questions from this chapter

In Exercises \(39-64,\) rationalize each denominator. $$\frac{5}{\sqrt[4]{x}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(\sqrt{x^{2}+9 x+3}=-x\) has no solution because a principal square root is always nonnegative.

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is \(\frac{w}{h}=\frac{2}{\sqrt{5}-1}\) The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (PICTURE NOT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

Let \(f(x)=\sqrt{9+x} .\) Find \(f(3 \sqrt{5}) \cdot f(-3 \sqrt{5})\)

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. $$\begin{aligned} &\sqrt{x}+4=2\\\ &[-2,18,1] \text { by }[0,10,1] \end{aligned}$$
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