Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\left( \frac{x^{-2} + y^{-2}}{x^{-2} - y^{-2}} \right) \times \left( \frac{x + y}{x - y} \right)\).
2. Recognize that \(x^{-2} = \frac{1}{x^2}\) and \(y^{-2} = \frac{1}{y^2}\). Rewrite the numerator and denominator of the first fraction using positive exponents: \(\frac{\frac{1}{x^2} + \frac{1}{y^2}}{\frac{1}{x^2} - \frac{1}{y^2}}\).
3. Find a common denominator for both the numerator and denominator of the large fraction: For the numerator, combine \(\frac{1}{x^2} + \frac{1}{y^2}\) as \(\frac{y^2 + x^2}{x^2 y^2}\); for the denominator, combine \(\frac{1}{x^2} - \frac{1}{y^2}\) as \(\frac{y^2 - x^2}{x^2 y^2}\).
4. Rewrite the large fraction as \(\frac{\frac{y^2 + x^2}{x^2 y^2}}{\frac{y^2 - x^2}{x^2 y^2}}\). Since both numerator and denominator share the same denominator \(x^2 y^2\), simplify by multiplying the numerator by the reciprocal of the denominator: \(\frac{y^2 + x^2}{x^2 y^2} \times \frac{x^2 y^2}{y^2 - x^2} = \frac{y^2 + x^2}{y^2 - x^2}\).
5. Now multiply this result by the second fraction \(\frac{x + y}{x - y}\): \(\frac{y^2 + x^2}{y^2 - x^2} \times \frac{x + y}{x - y}\). Notice that \(y^2 - x^2\) can be factored as \((y - x)(y + x)\). Use this factorization to simplify the expression further, and rewrite all terms with positive exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers, including negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive exponent, e.g., x^-2 = 1/x^2. Understanding these laws is essential to rewrite expressions with positive exponents.

Algebraic Fractions and Simplification

Algebraic fractions involve variables in numerator and denominator. Simplifying such expressions requires factoring, finding common denominators, and canceling common factors. Mastery of these techniques helps in reducing complex rational expressions to simpler forms.

Difference of Squares

The difference of squares is a factoring pattern: a^2 - b^2 = (a - b)(a + b). Recognizing this pattern allows factoring expressions like x^-2 - y^-2 by rewriting them as (1/x^2) - (1/y^2), which can be factored further. This is key to simplifying the given expression.