1.1 Let $a$ And $b$ Be Real Numbers Such That $ A \textless B A \ \textless \ B A \textless B [/tex]. With Reasons, Determine $\left|(b-a)^2(a-b)\right|$. Do Not Expand (multiply Out).

Mar 13, 2025 by ADMIN 193 views

**Understanding the Problem and Its Implications**

Problem Overview

We are given two real numbers, $a$ and $b$ , with the condition that $a < b$ . Our task is to determine the value of $\left|(b-a)^2(a-b)\right|$ without expanding the expression.

Analyzing the Expression

To begin, let's break down the given expression into its components. We have $(b-a)^2$ and $(a-b)$ . Since $a < b$ , we know that $b - a$ is a positive quantity. Therefore, $(b-a)^2$ is also positive.

Properties of Absolute Value

The absolute value of a number is its distance from zero on the number line. In this case, we have the expression $\left|(b-a)^2(a-b)\right|$ . Since $(b-a)^2$ is positive, we can simplify the expression as follows:

\left|(b-a)^2(a-b)\right| = \left|(b-a)^2\right| \cdot \left|(a-b)\right|

Simplifying the Expression

Now, let's simplify the expression further. We know that the absolute value of a positive number is the number itself. Therefore, we can simplify the expression as follows:

\left|(b-a)^2\right| \cdot \left|(a-b)\right| = (b-a)^2 \cdot \left|(a-b)\right|

Determining the Value of the Expression

Since $a < b$ , we know that $b - a$ is a positive quantity. Therefore, $(a-b)$ is a negative quantity. The absolute value of a negative number is its positive counterpart. Therefore, we can simplify the expression as follows:

(b-a)^2 \cdot \left|(a-b)\right| = (b-a)^2 \cdot -(a-b)

Final Simplification

Now, let's simplify the expression further. We can combine the terms as follows:

(b-a)^2 \cdot -(a-b) = -(b-a)^2(a-b)

Conclusion

Therefore, the value of $\left|(b-a)^2(a-b)\right|$ is $\boxed{-(b-a)^2(a-b)}$ .

Discussion and Implications

The given problem requires us to determine the value of an expression involving absolute value and real numbers. We used properties of absolute value and simplified the expression step by step to arrive at the final answer. This problem has implications in various areas of mathematics, including algebra and calculus.

Real-World Applications

The concept of absolute value and its properties have numerous real-world applications. For example, in finance, absolute value is used to calculate the difference between two values, such as the difference between the current stock price and the previous stock price. In physics, absolute value is used to calculate the distance between two points in space.

Future Research Directions

This problem has implications for future research in mathematics, particularly in the areas of algebra and calculus. Researchers can explore the properties of absolute value and its applications in various fields, such as finance and physics.

Conclusion

Q: What is the definition of absolute value?

A: The absolute value of a number is its distance from zero on the number line. It is denoted by the symbol | | and is always non-negative.

Q: How do you simplify an expression involving absolute value?

A: To simplify an expression involving absolute value, you need to follow these steps:

Identify the absolute value expression.
Determine the sign of the expression inside the absolute value.
If the expression is positive, the absolute value is the expression itself.
If the expression is negative, the absolute value is the negative of the expression.

Q: What is the difference between absolute value and modulus?

A: Absolute value and modulus are two terms that are often used interchangeably. However, in some contexts, modulus refers specifically to the absolute value of a complex number.