\( P=\left(\frac{1}{4}\right)^{-\left(\frac{1}{2}\right)^{-2}}+\left(\frac{1}{2}\right)^{-\left(\frac{1}{2}\right)^{-1}}+\left(\frac{1}{3}\right)^{-1} \) \( \begin{array}{lll}\text { a. } 250 & \text { c. } 263 & \text { e. } 275 \\ \text { b. } 258 & \text { d. } 270 & \end{array} \)

To solve for \( P \), let's break down the terms individually. First, simplifying the term \( \left(\frac{1}{4}\right)^{-\left(\frac{1}{2}\right)^{-2}} \): 1. Calculate \( \left(\frac{1}{2}\right)^{-2} = 2^{2} = 4 \). 2. Then evaluate \( \left(\frac{1}{4}\right)^{-4} = 4^{4} = 256 \). Next, simplifying the second term \( \left(\frac{1}{2}\right)^{-\left(\frac{1}{2}\right)^{-1}} \): 1. Compute \( \left(\frac{1}{2}\right)^{-1} = 2 \). 2. Now evaluate \( \left(\frac{1}{2}\right)^{-2} = 2^{2} = 4 \). Finally, simplifying the third term \( \left(\frac{1}{3}\right)^{-1} = 3 \). Now adding these results together: \[ P = 256 + 4 + 3 = 263. \] So, the answer is: \( \text{c. } 263 \). Now, let’s provide some additional knowledge! Understanding how exponents work is essential, especially with negative powers! Negative exponents imply that you’re taking the reciprocal of the base. For example, \( a^{-n} = \frac{1}{a^n} \). This becomes crucial when simplifying complex expressions, as it can turn what seems like a daunting calculation into manageable steps that rely on fundamental properties of exponents. In real-life applications, understanding exponential functions can help in areas like finance (calculating compound interest), physics (investigating decay rates), and even computer science (algorithm analysis). Exponential growth and decay models describe a plethora of phenomena, from population growth trends to the spread of diseases, making them an integral part of many scientific inquiries!