- Simplify \(\frac{1}{64}\) as \(4^{-3}\) and \(2^{-6}\).
- Evaluate each given expression using exponent laws.
- Identify expressions (c) and (e) as equal to \(\frac{1}{64}\).
- The correct expressions are \(\boxed{c, e}\).
Explanation
- Analysis of the Expressions
The objective is to determine which two of the five provided expressions evaluate to \(\frac{1}{64}\). First, we note that \(\frac{1}{64}\) can be written as \(\frac{1}{4^3}\) or \(\frac{1}{2^6}\). Using the exponent rules \(x^{-n} = \frac{1}{x^n}\) and \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\), we can simplify each given expression to check for equality with \(\frac{1}{64}\).
- Evaluating and Comparing the Expressions
We evaluate each of the five expressions one by one:
A\(\left(\frac{1}{4}\right)^{-3} = 4^3 = 64\)
B\(6^{-4} = \frac{1}{6^4} = \frac{1}{1296}\)
C\(4^{-3} = \frac{1}{4^3} = \frac{1}{64}\)
D\(\left(\frac{1}{2}\right)^{-5} = 2^5 = 32\)
E\(2^{-6} = \frac{1}{2^6} = \frac{1}{64}\)
By comparing these results, we find that the expressions (c) and (e) are equal to \(\frac{1}{64}\).
Examples
Exponents are used in many scientific and financial contexts. For example, in computer science, memory sizes are often powers of 2. If a system requires a specific fraction of a memory block, say \(2^{-6}\) of a total capacity, calculating this efficiently using exponent rules helps programmers optimize memory allocation and understand system constraints. Similarly, scientific notation uses negative exponents to represent very small measurements in physics and chemistry.