Perform all indicated operations, and write each answer with positive integer exponents. {x- 9y^-1}/{ (x-3y^-1)(x+3y^-1)}

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\frac{x - 9y^{-1}}{(x - 3y^{-1})(x + 3y^{-1})}\).
2. Recognize that the denominator is a product of two binomials in the form \((a - b)(a + b)\), which can be simplified using the difference of squares formula: \(a^2 - b^2\).
3. Apply the difference of squares formula to the denominator: \((x)^2 - (3y^{-1})^2 = x^2 - 9y^{-2}\).
4. Rewrite the numerator \(x - 9y^{-1}\) as it is, and express all terms with positive exponents by converting \(y^{-1}\) to \(\frac{1}{y}\) and \(y^{-2}\) to \(\frac{1}{y^2}\) if needed, but keep them in exponential form for now.
5. Now, look for common factors or ways to simplify the fraction \(\frac{x - 9y^{-1}}{x^2 - 9y^{-2}}\) further, possibly by factoring the numerator or rewriting terms to match the denominator's structure.

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, y^-1 equals 1/y. Understanding how to rewrite expressions with negative exponents as fractions is essential for simplifying and performing operations correctly.

Polynomial Factoring

Factoring involves expressing a polynomial as a product of simpler polynomials. Recognizing patterns such as the difference of squares helps simplify expressions, especially in denominators like (x - 3y^-1)(x + 3y^-1), which can be factored into x^2 - (3y^-1)^2.

Simplifying Rational Expressions

Simplifying rational expressions requires combining like terms, factoring, and canceling common factors in numerators and denominators. Writing the final answer with positive integer exponents involves converting negative exponents and ensuring the expression is fully simplified.