To solve the expression \( (2^3 \times 2^{-1})^2 \) and find its equivalent in standard form, let's break it down step-by-step. ### Step 1: Simplify the expression inside the parentheses The expression inside the parentheses is \( 2^3 \times 2^{-1} \). When multiplying powers with the same base, we add the exponents. This is based on the exponent rule: \[ a^m \times a^n = a^{m+n} \] Applying this rule here: \[ 2^3 \times 2^{-1} = 2^{3 + (-1)} = 2^{3 - 1} = 2^2 \] ### Step 2: Raise the result to the power of 2 Now we take the result from Step 1, which is \( 2^2 \), and raise it to the power of 2: \[ (2^2)^2 \] When raising a power to another power, we multiply the exponents. This is another exponent rule: \[ (a^m)^n = a^{m \times n} \] Applying this rule: \[ (2^2)^2 = 2^{2 \times 2} = 2^4 \] ### Final Answer Thus, the expression \( (2^3 \times 2^{-1})^2 \) simplifies to \( 2^4 \). ### Correct Option The correct option is **A. \( 2^4 \)**. ### Explanation of Other Options - **B. \( 2^6 \)**: This option is incorrect because it suggests that the exponent should be 6. However, we calculated the exponent to be 4, not 6. - **C. \( 2^{10} \)**: This option is also incorrect. It implies a much larger exponent than what we derived. The calculations show that the exponent is 4, not 10. - **D. \( 2^2 \)**: This option is incorrect as well. While \( 2^2 \) was an intermediate result, it was not the final answer after raising it to the power of 2. ### Common Pitfalls 1. **Forgetting to add exponents**: When multiplying powers, always remember to add the exponents. 2. **Misapplying exponent rules**: Ensure you apply the correct rules for multiplying and raising powers. 3. **Not simplifying step-by-step**: It’s important to simplify expressions step-by-step to avoid confusion. ### Revision Summary - When multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \). - When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \). - The expression \( (2^3 \times 2^{-1})^2 \) simplifies to \( 2^4 \). - The correct answer is **A. \( 2^4 \)**.