- Simplify \((2^5)^{-2}\) using the rule \((a^m)^n = a^{m \cdot n}\) to obtain \(2^{-10}\).
- Convert \(2^{-10}\) to a fraction using \(a^{-m} = \frac{1}{a^m}\) to get \(\frac{1}{2^{10}}\).
- Calculate \(2^{10} = 1024\) to find the fraction \(\frac{1}{1024}\).
- Identify that the equivalent expressions are \(2^{-10}\) and \(\frac{1}{1024}\), which correspond to the second choice.
- Final Answer: \(\boxed{2^{-10} \text{ and } \frac{1}{1024}}\)
Explanation
- Simplifying the Expression Using Exponent Rules
The given expression is \((2^5)^{-2}\). To simplify this, we use the exponent power rule \((a^m)^n = a^{m \cdot n}\), where \(a=2\), \(m=5\), and \(n=-2\). Applying this rule, we multiply the exponents: \(5 \times (-2) = -10\). Thus, the expression simplifies to \(2^{-10}\).
- Converting to Fractional Form
Next, we convert the expression with a negative exponent into its fractional form using the rule \(a^{-m} = \frac{1}{a^m}\). Applying this rule to \(2^{-10}\), we get \(\frac{1}{2^{10}}\). Now, we calculate the value of \(2^{10}\). We know that \(2^5 = 32\), so \(2^{10} = 2^5 \times 2^5 = 32 \times 32 = 1024\). Therefore, the fractional equivalent of \(2^{-10}\) is \(\frac{1}{1024}\).
- Evaluating and Comparing with Options
Comparing our derived results, \(2^{-10}\) and \(\frac{1}{1024}\), with the provided options, we see that they match the second option. The other options are incorrect: \(2^{-10}\) is not \(\frac{1}{20}\), \(10^{-2}\) is \(\frac{1}{100}\), and \(10^{-10}\) is not \(\frac{1}{100}\). Thus, the correct equivalent pair is \(2^{-10}\) and \(\frac{1}{1024}\).
Examples
Understanding exponent rules is essential in fields like computer science and data storage. For example, when measuring memory sizes in bytes, computers use powers of 2. If a specific system operation involves reducing a memory allocation by a power of 2, understanding expressions like \((2^5)^{-2}\) helps engineers calculate the exact remaining capacity. This math skill allows for efficient resource management and better hardware utilization in real-world technology.