- Combine the terms in the numerator to a common denominator: \(\frac{y-2x^2}{x^2y}\).
- Express the entire expression as a fraction of two fractions: \(\frac{\frac{y-2x^2}{x^2y}}{\frac{y-2x^2}{1}}\).
- Multiply by the reciprocal to divide the fractions: \(\frac{y-2x^2}{x^2y} \times \frac{1}{y-2x^2}\).
- Simplify by canceling the common factor \((y-2x^2)\) to reach the final result: \(\boxed{\frac{1}{x^2y}}\).
Explanation
- Analyzing the Expression and Establishing the Setup
To simplify the expression \(\frac{\frac{1}{x^2}-\frac{2}{y}}{y-2x^2}\), we analyze the structure given in the steps provided.
Step 1 establishes a common denominator for the terms in the numerator:
\[\frac{1}{x^2} - \frac{2}{y} = \frac{y}{x^2y} - \frac{2x^2}{x^2y} = \frac{y - 2x^2}{x^2y}.\]
This brings our expression into the form \(\frac{\frac{y - 2x^2}{x^2y}}{y - 2x^2}\).
Step 2 expresses both the numerator and the denominator as fractions:
\[\frac{\frac{y - 2x^2}{x^2y}}{\frac{y - 2x^2}{1}}.\]
We assume \(x
eq 0\), \(y
eq 0\), and \(y - 2x^2
eq 0\) for the expression to be well-defined.
- Performing the Division and Simplifying
To divide one fraction by another, we multiply the first fraction by the reciprocal of the second:
\[\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}.\]
Substituting our terms \(A = y - 2x^2\), \(B = x^2y\), \(C = y - 2x^2\), and \(D = 1\), we have:
\[\frac{y - 2x^2}{x^2y} \cdot \frac{1}{y - 2x^2}.\]
Since \(y - 2x^2
eq 0\), we can cancel the common factor \((y - 2x^2)\) from the numerator and the denominator.
This leaves us with:
\[\frac{1}{x^2y}.\]
Thus, the expression simplifies to \(\boxed{\frac{1}{x^2y}}\).
Examples
In physics, calculating ratios of physical quantities is common. For instance, if you are determining the density of a complex fluid flow where you have a ratio of pressure differentials (\(\frac{1}{x^2}-\frac{2}{y}\)) relative to an internal drag force (\(y-2x^2\)), simplifying the expression allows you to isolate the variables \(x\) and \(y\) more effectively. Algebraic simplification is a fundamental tool for reducing complex formulas into manageable, functional expressions.