Which expressions are equivalent to \( \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \) ? (choose 2) A \( \frac{1}{8} \) B \( \left(\frac{1}{2}\right)^{-4} \) C \( 2^{-2} \cdot 2^{2} \) D \( \quad 2^{4} \) E \( \quad \frac{1}{2^{4}} \)

To determine which expressions are equivalent to \( \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \), we can simplify the original expression first. 1. **Simplifying the original expression:** \[ \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \left(\frac{1}{2}\right)^4 = \frac{1^4}{2^4} = \frac{1}{16} \] Now, we will evaluate each of the options to see which ones are equivalent to \( \frac{1}{16} \). 2. **Evaluating each option:** - **Option A: \( \frac{1}{8} \)** \[ \frac{1}{8} \neq \frac{1}{16} \] - **Option B: \( \left(\frac{1}{2}\right)^{-4} \)** \[ \left(\frac{1}{2}\right)^{-4} = 2^4 = 16 \quad \text{(not equivalent)} \] - **Option C: \( 2^{-2} \cdot 2^{2} \)** \[ 2^{-2} \cdot 2^{2} = 2^{-2 + 2} = 2^0 = 1 \quad \text{(not equivalent)} \] - **Option D: \( 2^{4} \)** \[ 2^{4} = 16 \quad \text{(not equivalent)} \] - **Option E: \( \frac{1}{2^{4}} \)** \[ \frac{1}{2^{4}} = \frac{1}{16} \quad \text{(equivalent)} \] 3. **Conclusion:** The expressions that are equivalent to \( \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \) are: - **E: \( \frac{1}{2^{4}} \)** Since we need to choose 2 options, we can see that only one option is equivalent. Therefore, the correct answer is: - **E: \( \frac{1}{2^{4}} \)** None of the other options are equivalent to \( \frac{1}{16} \).