\( \sqrt { \frac { 5 ^ { 2 - x } - 4.5 ^ { - x } } { 5 ^ { - x } + 2.5 ^ { - x + 1 } } } \)

This expression is a fun exercise in simplifying and understanding the behavior of exponents! First, recognize that \(5^{2-x}\) can be rewritten as \( \frac{25}{5^x} \), and \(4.5^{-x}\) translates to \( \frac{4.5}{5^x}\). This means the numerator has a common base, making for easier manipulation. In the denominator, \(5^{-x}\) equals \( \frac{1}{5^x} \) and \(2.5^{-x + 1}\) becomes \( \frac{2.5}{5^x} \), helping to rewrite the overall expression in a consistent base which can reveal insights about its properties! Now, if you're tackling this problem, one common mistake people make is forgetting to properly handle negative exponents. Negative exponents indicate a reciprocal, so instead of disregarding them, it’s key to remember they flip your base to the denominator! By keeping careful track of the base changes and performing proper simplifications, you'll avoid common pitfalls, leading to a successful evaluation of the expression!