- Identify the target value as \(\frac{1}{64} = 4^{-3} = 2^{-6}\).
- Evaluate each expression using exponent rules: \(\left(\frac{1}{4}\right)^{-3} = 64\), \(6^{-4} = \frac{1}{1296}\), \(4^{-3} = \frac{1}{64}\), \(\left(\frac{1}{2}\right)^{-5} = 32\), and \(2^{-6} = \frac{1}{64}\).
- Match the evaluated results with the target value.
- The correct expressions are: \(\boxed{c, e}\).
Explanation
- Analysis of the Problem
We need to determine which two of the five given expressions evaluate to the target value of \(\frac{1}{64}\). The expressions are:
A\(\left(\frac{1}{4}\right)^{-3}\)
B\(6^{-4}\)
C\(4^{-3}\)
D\(\left(\frac{1}{2}\right)^{-5}\)
E\(2^{-6}\)
To solve this, we will use the following fundamental properties of exponents:
- Negative exponent rule: \(x^{-n} = \frac{1}{x^n}\)
- Fraction to negative exponent rule: \(\left(\frac{1}{x}\right)^{-n} = x^n\)
- Powers of 2 and 4: \(64 = 4^3 = 2^6\). Thus, \(\frac{1}{64} = \frac{1}{4^3} = \frac{1}{2^6}\).
- Evaluation of Expressions
We evaluate each expression systematically:
- Expression a: \(\left(\frac{1}{4}\right)^{-3} = 4^3 = 64\). This is not equal to \(\frac{1}{64}\).
- Expression b: \(6^{-4} = \frac{1}{6^4} = \frac{1}{1296}\). This is not equal to \(\frac{1}{64}\).
- Expression c: \(4^{-3} = \frac{1}{4^3} = \frac{1}{64}\). This matches the target value.
- Expression d: \(\left(\frac{1}{2}\right)^{-5} = 2^5 = 32\). This is not equal to \(\frac{1}{64}\).
- Expression e: \(2^{-6} = \frac{1}{2^6} = \frac{1}{64}\). This matches the target value.
- Conclusion
Comparing the results calculated for each expression with our target of \(\frac{1}{64}\), we find that expression (c) and expression (e) both evaluate to exactly \(\frac{1}{64}\).
Thus, the expressions that Iris could have evaluated are (c) and (e). The final answer is \(\boxed{c, e}\).
Examples
Understanding exponent rules is essential for scaling in real-life applications. For example, if you are designing a fractal structure or managing digital storage, you might work with powers of 2 (like bytes in computer memory). Recognizing that \(\frac{1}{64}\) can be written as \(4^{-3}\) or \(2^{-6}\) allows you to simplify complex calculations and optimize storage or design, making it easier to handle data representations efficiently.