Solving and Simplifying Complex Algebraic Expressions: A Step-by-Step Guide

Algebraic expressions can often appear complex due to their intricate design. Understanding how to navigate and solve these expressions is crucial for anyone working with mathematics. In this article, we’ll break down and simplify the given algebraic expression: (frac{x^2 cdot frac{1}{x-1}^2}{x} 2xy 2). We’ll walk through the process step-by-step, explaining each critical step involved in solving the expression.

Introduction to the Expression

The given expression is: (frac{x^2}{(x-1)^2}x^2 2xy 2). To fully understand and simplify this expression, we first need to clarify the meaning of (frac{1}{(x-1)^2}) and the overall expression structure.

Simplifying the Expression

The expression can be simplified into a format that is more manageable. Here’s how we can break it down:

x^2 / (x-1)^2 * x^2   2xy   2

Step 1: Rewrite the Expression

First, let's rewrite the expression:

x^2 / (x-1)^2 * x^2   2xy   2

Step 2: Combine the Terms

Next, we combine the terms by expressing (x^2) with a common denominator:

x^2  (x^2) / (x-1)^2

Now, we can rewrite the expression as:

(x^2   x^2 * 2xy   2) / (x-1)^2

Simplify further by distributing the terms:

x^4 - 2x^3   2x^2 - 2xy   2 / (x-1)^2

Step 3: Expand and Combine the Numerator

Expanding and combining the terms in the numerator:

x^4 - 2x^3   2x^2   2xy   2

We now have the simplified expression:

(x^4 - 2x^3   2x^2   2xy   2) / (x-1)^2

Example with Specific Values

For better understanding, let’s plug in specific values (x 2) and (y 3):

2^2 / (2-1)^2 * 2^2   2*2*3   2

Step by step, we solve:

4 / 1^2 * 4   12   2

1. Simplify the denominator:

1  1

2. Simplify the product:

4 * 4  16

3. Combine the terms:

16   12   2  30

4. Final expression:

16 / 1  16

Thus, when (x 2) and (y 3), the expression simplifies to 30.

Conclusion

This example demonstrates the process of simplifying and solving complex algebraic expressions. To get a better understanding and to solve expressions confidently, it’s essential to practice with different values and to verify each step. If you encounter any issues or have further questions, feel free to comment or reach out for assistance.