- The pattern for negative powers of 2 is \(2^{-n} = \frac{1}{2^n}\).
- Applying this to \(n=6\), we get \(2^{-6} = \frac{1}{2^6}\).
- Calculating \(2^6\) yields \(64\).
- Thus, \(a = \frac{1}{64}\), which matches option c.
- The final answer is \(\boxed{\frac{1}{64}}\).
Explanation
- Analyzing the Pattern
The problem presents a pattern of powers of 2 with negative exponents. We are given the following:
\(2^{-1} = \frac{1}{2}\)
\(2^{-2} = \frac{1}{4}\)
\(2^{-3} = \frac{1}{8}\)
\(2^{-4} = \frac{1}{16}\)
\(2^{-5} = \frac{1}{32}\)
The pattern suggests that for any positive integer \(n\), the expression \(2^{-n}\) is equivalent to \(\frac{1}{2^n}\). Our objective is to find the value of \(a\) where \(a = 2^{-6}\) based on this established rule.
- Calculating the Value of \(a\)
Using the identified rule \(2^{-n} = \frac{1}{2^n}\), we substitute \(n = 6\) into the expression:
\(2^{-6} = \frac{1}{2^6}\).
Now, we calculate the value of \(2^6\) by multiplying 2 by itself six times:
\(2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\).
Step-by-step:
\(2 \times 2 = 4\)
\(4 \times 2 = 8\)
\(8 \times 2 = 16\)
\(16 \times 2 = 32\)
\(32 \times 2 = 64\)
Therefore, \(2^6 = 64\), which leads us to \(a = \frac{1}{64}\).
- Final Conclusion
Comparing our result \(a = \frac{1}{64}\) to the provided options:
A\(-64\)
B\(\frac{1}{16}\)
C\(\frac{1}{64}\)
The calculated value corresponds to option c. Thus, the value of \(a\) is \(\frac{1}{64}\).
Examples
Understanding powers of 2 with negative exponents is fundamental in fields like computer science, where memory addressing and data storage capacities are often measured in powers of 2. For instance, if a system uses a specific bit depth for indexing, a decrease in the exponent (like moving from \(2^{-5}\) to \(2^{-6}\)) represents a finer level of granularity in data precision. Mastering this pattern helps in managing digital resources efficiently and understanding how bit-level operations scale.