Find All Possible Integer Solutions Of The Equation:$\[ |x|^{x \cdot (2x-3)} - \left(x^2\right)^{(x-1)} \cdot (x-2) \cdot |x|^{2018} = 0 \\]

Introduction

The given equation is a complex mathematical expression involving absolute values, exponents, and polynomials. The goal is to find all possible integer solutions for the variable $x$ that satisfy the equation. In this discussion, we will break down the equation, analyze its components, and use mathematical techniques to find the solutions.

Breaking Down the Equation

The given equation can be broken down into three main components:

1. $|x|^{x \cdot (2x-3)}$
2. $\left(x^2\right)^{(x-1)}$
3. $(x-2) \cdot |x|^{2018}$

Each component will be analyzed separately to understand its behavior and how it contributes to the overall equation.

Component 1: $|x|^{x \cdot (2x-3)}$

The first component involves the absolute value of $x$ raised to the power of $x \cdot (2x-3)$. Since the absolute value of $x$ is always non-negative, the exponent $x \cdot (2x-3)$ will determine the behavior of this component.

To find the possible values of $x$ that make this component equal to zero, we need to consider the cases when the exponent is equal to zero.

• If $x \cdot (2x-3) = 0$, then either $x = 0$ or $2x-3 = 0$. Solving for $x$ in the second equation gives $x = \frac{3}{2}$, which is not an integer solution. Therefore, the only possible integer solution for this component is $x = 0$.

Component 2: $\left(x^2\right)^{(x-1)}$

The second component involves the square of $x$ raised to the power of $x-1$. Since the square of $x$ is always non-negative, the exponent $x-1$ will determine the behavior of this component.

To find the possible values of $x$ that make this component equal to zero, we need to consider the cases when the exponent is equal to zero.

• If $x-1 = 0$, then $x = 1$. This is a possible integer solution for this component.

Component 3: $(x-2) \cdot |x|^{2018}$

The third component involves the product of $(x-2)$ and $|x|^{2018}$. Since the absolute value of $x$ is always non-negative, the exponent $2018$ will determine the behavior of this component.

To find the possible values of $x$ that make this component equal to zero, we need to consider the cases when either $(x-2)$ or $|x|^{2018}$ is equal to zero.

• If $(x-2) = 0$, then $x = 2$. This is a possible integer solution for this component.
• If $|x|^{2018} = 0$, then $x = 0$. This is also a possible integer solution for this component.

Combining the Components

Now that we have analyzed each component separately, we can combine them to find the possible integer solutions for the entire equation.

From the analysis of the components, we have found the following possible integer solutions:

• $x = 0$
• $x = 1$
• $x = 2$

These solutions satisfy the equation and are the only possible integer solutions.

Conclusion

In this discussion, we have analyzed the given equation and broken it down into three main components. We have used mathematical techniques to find the possible integer solutions for each component and combined them to find the overall solutions for the equation. The possible integer solutions are $x = 0$, $x = 1$, and $x = 2$. These solutions satisfy the equation and are the only possible integer solutions.

Final Answer

The final answer is $\boxed{0, 1, 2}$.

Introduction

In our previous discussion, we analyzed the given equation and found the possible integer solutions. In this Q&A article, we will address some common questions and provide additional insights into the solution.

Q: What is the significance of the absolute value in the equation?

A: The absolute value in the equation is crucial in determining the behavior of the components. It ensures that the exponent is always non-negative, which affects the possible values of $x$ that satisfy the equation.

Q: How did you find the possible integer solutions for each component?

A: We used mathematical techniques to analyze each component separately. For the first component, we considered the cases when the exponent is equal to zero. For the second component, we looked for the values of $x$ that make the exponent equal to zero. For the third component, we considered the cases when either $(x-2)$ or $|x|^{2018}$ is equal to zero.

Q: Why are the solutions $x = 0$, $x = 1$, and $x = 2$ the only possible integer solutions?

A: These solutions satisfy the equation and are the only possible integer solutions because they are the only values of $x$ that make each component equal to zero. For $x = 0$, the first component is equal to zero because the exponent is equal to zero. For $x = 1$, the second component is equal to zero because the exponent is equal to zero. For $x = 2$, the third component is equal to zero because $(x-2)$ is equal to zero.

Q: Can you provide more examples of equations that involve absolute values and exponents?

A: Yes, here are a few examples:

• $|x|^{x^2} - (x-1) \cdot |x|^{2019} = 0$
• $|x|^{x \cdot (x-1)} - (x^2) \cdot (x-2) \cdot |x|^{2018} = 0$
• $|x|^{x^3} - (x-1) \cdot (x-2) \cdot |x|^{2019} = 0$

These equations involve absolute values and exponents, and can be solved using similar techniques.

Q: How can I apply the techniques used in this solution to other mathematical problems?

A: The techniques used in this solution can be applied to other mathematical problems that involve absolute values and exponents. By analyzing each component separately and considering the cases when the exponent is equal to zero, you can find the possible values of the variable that satisfy the equation.

Conclusion

In this Q&A article, we have addressed some common questions and provided additional insights into the solution. We have also provided examples of equations that involve absolute values and exponents, and discussed how to apply the techniques used in this solution to other mathematical problems.

Final Answer

The final answer is $\boxed{0, 1, 2}$.