Simplifying Mathematical Expressions: An In-Depth Guide

Mathematics often involves manipulating and simplifying expressions to make them more manageable for further calculations or analysis. This guide will focus on a specific example: 1/x^2 - 2/x 3x. We will explore the process of simplifying such an expression, discuss common denominators, and walk through the steps involved.

Understanding the Expression

The given expression is 1/x^2 - 2/x 3x. This expression involves variables and fractions, and the goal is to simplify it while maintaining its mathematical equivalence. It is important to note that this expression is undefined when x 0, as division by zero is not allowed in mathematics.

Common Denominator

The first step in simplifying the expression is to identify and use a common denominator. In the given expression, the denominators are x^2 and x. The least common denominator (LCD) for these denominators is x^2. We will multiply each term by a form of 1 that will result in the common denominator:

1/x^2 * x/x 2/x * x^2/x * 3x

This results in:

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

Which simplifies to:

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

Multiplying by a Common Term

To further simplify, we can multiply the entire expression by x^2/x^2, which is equivalent to 1 and thus does not change the value of the expression:

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

This results in:

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

After simplification, this becomes:

1 - 2x * 3x^3

Which further simplifies to:

1 - 6x^4

And the final form is:

1 - 6x^4

Final Expression

After simplifying the expression, we obtain:

(1 - 6x^4) / x^2

This is the simplified form of the original expression. The expression is now easier to work with for further mathematical operations or analysis.

Conclusion

In conclusion, simplifying mathematical expressions involves identifying common denominators, using multiplication by a form of 1, and rearranging terms. By following these steps, complex expressions like 1/x^2 - 2/x 3x can be simplified to a more manageable form.

Keywords: simplifying expressions, algebraic simplification, common denominator