Solving the Expression x1/x - x-1/x: A Comprehensive Guide
When dealing with mathematical expressions, clarity is crucial. Often, expressions can be ambiguous or misinterpreted due to the complexity or lack of proper notation. This article will guide you through the process of simplifying and solving the expression x1/x - x-1/x. We will explore the underlying algebraic principles and provide a step-by-step solution.
Introduction to Mathematical Expressions
Mathematical expressions can sometimes be ambiguous when written in a single-line form. To avoid confusion, using clear notation such as parentheses or fractions is essential. The expression x1/x - x-1/x can be interpreted in two main ways:
(x1/x) - (x-1/x) (x1/x - x) / xBoth interpretations will be addressed in this article, provided with detailed explanations and step-by-step solutions.
Simplifying the Expression (x1/x) - (x-1/x)
Step 1: Understanding the Expression
The first interpretation of the expression is (x1/x) - (x-1/x). This expression consists of two parts: x1/x and x-1/x. These parts are subtracted from each other.
Step 2: Simplify Using Algebraic Identities
To simplify the expression x1/x - x-1/x, we can use the difference of squares identity: a2 - b2 (a - b)(a b). In this case, let:
a x1/x b x-1/xApplying the difference of squares identity, we get:
(x1/x)2 - (x-1/x)2 (x1/x - x-1/x)(x1/x x-1/x)
The term (x1/x x-1/x) is the sum of the two terms in the original expression.
Step 3: Final Simplified Expression
Therefore, the simplified expression is:
x2 -1/x2
Solving the Alternative Expression (x1/x - x) / x
Step 1: Understanding the Alternative Expression
For the second interpretation, we have the expression (x1/x - x) / x. This is a division involving a difference of terms.
Step 2: Simplifying the Numerator
The numerator is x1/x - x. We need to divide this by the denominator x.
The numerator can be rewritten as:
x1/x - x x1/x - x1
Step 3: Dividing Each Term Separately
Divide each term in the numerator by the denominator:
(x1/x - x1) / x x-1/x - 1
This simplifies the original expression to:
(x1/x - x) / x x-1/x - 1
Conclusion
The two possible solutions for the expression x1/x - x-1/x are:
x2 - 1/x2 x-1/x - 1Understanding and applying algebraic identities, such as the difference of squares, can greatly help in simplifying and solving complex expressions. It is important to ensure that the expressions are clearly written to avoid any ambiguity.