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When you see a negative exponent, don't let it intimidate you. It simply means you're working with the reciprocal of the base. For example, if you have \(a^{-n}\), this is the same as \(\frac{1}{a^n}\). In simple terms, flip the base (turn it upside down) and remove the negative in the exponent. In our exercise, this meant turning \(\left(\frac{5}{6}\right)^{-1.3}\) into \(\left(\frac{6}{5}\right)^{1.3}\). This approach simplifies the calculation and turns a negative exponent into a positive one, making it easier to work with and eventually evaluate using a calculator.

Fractional exponents are an elegant way to express roots and powers in a compact form. For instance, \(a^{1/n}\) indicates the \(n\)-th root of \(a\). In the case of more complex expressions, such as \(a^{m/n}\), it combines both rooting and powering. It can be understood as \((a^{1/n})^m\), which means first taking the root and then raising the result to a power. In our example, we computed \(1.2^{1.3}\), which involves both root and power operations done in sequence. By using fractional exponents, calculations become streamlined and often more manageable, especially when used in advanced mathematical operations.

When using calculators for evaluating powers and roots, a few steps can make the process smooth. Start by locating and familiarizing yourself with the exponent function. It might be labeled as '^', 'power', or similar buttons. After identifying this, enter the base number, followed by the exponent symbol, and then input the exponent value. For our example, once you enter \(1.2\), using the power function to input \(1.3\) results in the correct calculation of \(1.2^{1.3}\). Make sure to carefully follow each button press and type the numbers precisely to ensure accuracy. Depending on your calculator's precision, results might slightly vary, but typically, modern calculators will show several decimal places for precision in your displayed solution.