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Chapter 2: Problem 63

\(z^{2}=z^{2}\)

Short Answer

Expert verified
Any value of \(z\) satisfies the equation.

Step by step solution

01

Understand the Equation

Recognize that the given equation is an identity since both sides of the equation are exactly the same: \(z^2 = z^2\).
02

Simplify the Equation

Since the equation is already simplified and balanced (the left side is identical to the right side), no further simplification is required.
03

Determine the Values of z

The equation \(z^2 = z^2\) holds true for any value of \(z\). This means that \(z\) can be any real or complex number.

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Key Concepts

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

Identity in Algebra
In algebra, an identity is an equation that is true for all values of the variable involved. This means that no matter what value you substitute for the variable, the equation will always hold true.

In the given exercise, the equation is represented as \(z^2 = z^2\). This is a clear example of an identity because both sides of the equation are exactly the same.

Identities are useful because they help us recognize when an equation does not need further solving, and they often simplify the process of working with more complex equations.
Simplification of Equations
Simplifying equations is a fundamental skill in algebra. It involves reducing an equation to its simplest form while maintaining its equality. This helps make the equation easier to solve or understand.

In our given example, the equation \(z^2 = z^2\) is already in its simplest form. There’s nothing to change or reduce because both sides are identical.

This demonstrates an important point: not all equations need to be simplified further. Recognizing when an equation is already simplified is a valuable skill because it saves time and effort.
Value Determination in Mathematics
Determining the values of a variable that satisfy an equation is a key task in algebra. It involves finding the values that make the equation true.

In the context of the provided equation \(z^2 = z^2\), since the equation is an identity, it holds true for all values of \(z\). This means \(z\) can be any real or complex number, making value determination straightforward here.

For other types of equations, however, determining the values often requires solving the equation systematically. Techniques may include factoring, using the quadratic formula, or employing other algebraic methods.

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Most popular questions from this chapter

Find each of the following in the \(x+i y\) form. $$ \cosh (i+2) $$

Find each of the following in the \(x+i y\) form. $$ \sinh \left(1+\frac{1 \pi}{2}\right) $$

In each of the following problems, \(z\) represents the displacement of a particle from the origin. Find (as functions of \(t)\) its speed and the magnitude of its acceleration, and describe the motion. $$ z=(1+i) e^{i t} $$

Find all the values of the indicated roots and plot them. $$ \sqrt[3]{27} $$

\(5\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right)\)
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