91Ó°ÊÓ

Exponents are used to represent repeated multiplication. For example, when we say \(2^3\), this means \(2 \times 2 \times 2\), which equals 8. In the case of the exercise, we encounter a more advanced use of exponents, where negative exponents are applied.

Negative exponents indicate the inverse of the base raised to the positive exponent. Thus, \(x^{-n}\) means 'one divided by (x raised to the n)'. In other words, if we have \(9^{-2}\), this is the same as \(1/9^2\) or \(1/81\).

This might seem a bit complex at first, but remember that the negative sign doesn't mean the number is negative. It simply instructs us to take the inverse (or reciprocal) of the base number raised to the positive power.

Inverse operations are mathematical operations that undo each other, such as addition and subtraction or multiplication and division. When dealing with exponents, the inverse operation is taking a root or using a reciprocal.

In our exercise, the negative power is showing us the reciprocal operation. Instead of multiplying the base number by itself, we are dividing 1 by the base number raised to a positive power. So when we're trying to solve for \(3^{-3}\), we're actually finding the reciprocal of \(3^3\), which is \(1/3^3\) or \(1/27\).

Note that understanding inverse operations is crucial in algebra because it helps us solve for unknowns and simplify expressions, two key skills in advancing mathematical comprehension.

Simplifying expressions means to rewrite them in a more concise and easier to understand form without changing their value. It often involves combining like terms, using inverse operations, and applying exponent rules.

In the context of our exercise, simplification involves evaluating negative exponents to produce fractions, which are then added together. We converted the expressions with negative exponents into their positive counterparts in fractional form, resulting in \(1/81\) for \(9^{-2}\) and \(1/27\) for \(3^{-3}\). Afterward, we added these fractions to find the simplified answer to the problem.

Remember, simplifying expressions makes complex problems more manageable and thus is a fundamental aspect of higher-level math. Always double-check to ensure that every possible simplification has been made, while ensuring the original expression's value remains unchanged.