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Chapter 2: Problem 80

Simplify the complex number and write it in standard form. \(4 i^{2}-2 i^{3}\)

Short Answer

Expert verified
The simplified complex number in standard form is -4 + 2i.

Step by step solution

01

Analyzing the properties of i

The imaginary unit i is represented in such a way that \(i^{2} = -1\) and \(i^{3} = -i\). This characteristics will be used in the next steps.
02

Substitution

Substitute \(i^{2}\) and \(i^{3}\) with their corresponding values. The expression becomes \(4 \cdot -1 - 2 \cdot -i\).
03

Simplification

Simplify the obtained expression to get -4 + 2i.

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Key Concepts

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

Imaginary Unit
Complex numbers often include an imaginary component, denoted by the letter "i." This imaginary unit is not just a fanciful concept but a core building block in complex number arithmetic.
When we square the imaginary unit, the result is -1, represented as \( i^2 = -1 \). This relationship allows for interesting interactions when powers of "i" are further considered.
  • \( i^1 = i \) is the first power and remains unchanged.
  • \( i^2 = -1 \) reveals the key property of the imaginary unit.
  • \( i^3 = i^2 \times i = -1 \times i = -i \) flips the sign of "i."
  • \( i^4 = i^3 \times i = -i \times i = 1 \)
These properties repeat cyclically every four powers. Understanding this cycle helps in simplifying expressions like \( i^{72} \) by recognizing patterns.
Standard Form
Every complex number has a real part and an imaginary part. The standard form of a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers.
In this form:
  • "a" represents the real component.
  • "bi" is the imaginary component.
For example, in the number -4 + 2i, -4 is the real part and 2i is the imaginary part.
Writing numbers in standard form not only standardizes how complex numbers are noted but also sets the stage for more advanced mathematical operations. An expression such as \( 4i^2 - 2i^3 \), once simplified, adheres to this form seamlessly, showing the clarity and efficiency of using the standard form.
Simplification
Simplification in the realm of complex numbers involves reducing expressions to their basal form. This includes substituting powers of the imaginary unit and combining like terms.
Here's the sequence to simplify \( 4i^2 - 2i^3 \):
  • Begin with recognizing the powers: \( i^2 = -1 \) and \( i^3 = -i \).
  • Substitute these values into the expression: \( 4(-1) - 2(-i) \).
  • Calculate the operations: \( -4 + 2i \).
Each step in the simplification process is straightforward. Use properties of "i" and algebraic rules to simplify complex expressions effectively. It's akin to solving a puzzle, where understanding each piece's role leads to a satisfying resolution.

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Most popular questions from this chapter

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

The revenue and cost equations for a product are \(R=x(50-0.0002 x)\) and \(C=12 x+150,000\) where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold. How many units must be sold to obtain a profit of at least \(\$ 1,650,000 ?\) What is the price per unit?

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. \(x^{2} \leq 16\)

(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. \(x^{2}+b x+4=0\)
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