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When you see a negative exponent in mathematics, it doesn’t mean that the value itself is negative. Instead, a negative exponent indicates that we take the reciprocal of the base and then raise it to the positive of that exponent. For example, when you have an expression like \(3^{-2}\), it can be rewritten as \(\frac{1}{3^2}\). Essentially, you are saying: "Let’s flip it over, and then square it". Thus, \(3^{-2}\) becomes \(\frac{1}{9}\), and \(3^{-1}\) becomes \(\frac{1}{3}\). This process allows you to change negative exponents into positive ones while maintaining the value of the expression.
Understanding this concept is a fundamental skill, as it often appears in algebra and beyond:

• A negative exponent flips the base to the denominator.
• Calculate the positive exponent of the reciprocal of the base.
• Always express your final answers with positive exponents.

When you are dealing with fractions and need to add or subtract them, finding a common denominator is an essential step. A common denominator is a shared multiple of the denominators in the fractions you’re working with. By converting fractions to have a common denominator, you can easily perform addition or subtraction on them.
In our case, to add \(\frac{1}{9}\) and \(\frac{1}{3}\), we look for the smallest common multiple of 9 and 3, which is 9 itself. Once you have the common denominator (9), you simply adjust the fractions if necessary:

• \(\frac{1}{9}\) is already correct.
• Rewrite \(\frac{1}{3}\) as \(\frac{3}{9}\) to match the common denominator.
• You can now add the numerators: \(\frac{1}{9} + \frac{3}{9} = \frac{4}{9}\).

This method provides a straightforward pathway to adding and simplifying fractions.

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying them involves a process similar to arithmetic fractions, including finding common denominators and converting negative exponents. The exercise we tackled, although simple, featured rational expressions with integers and negative exponents, which are a common type of rational expression.
The key steps to work with rational expressions include:

• Simplifying the fractions by reviewing all exponents.
• Ensuring all terms have a common denominator if you are performing addition or subtraction.
• In case of negative exponents, convert them to positive by taking reciprocals.

As you become more comfortable with these steps, handling more complex rational expressions becomes easier. Besides simplifying, these skills extend to solving equations and further algebraic manipulations.