To simplify the expression \( (2^3 \times 2^{-1})^2 \) and express the answer in standard form, let's break it down step by step. ### Step 1: Simplify the Inside of the Parentheses The expression inside the parentheses is \( 2^3 \times 2^{-1} \). When multiplying numbers with the same base, we can use the property of exponents that states: \[ a^m \times a^n = a^{m+n} \] In our case, we have: \[ 2^3 \times 2^{-1} = 2^{3 + (-1)} = 2^{3 - 1} = 2^2 \] ### Step 2: Raise the Result to the Power of 2 Now we need to raise \( 2^2 \) to the power of 2. Again, we can use the property of exponents: \[ (a^m)^n = a^{m \times n} \] Applying this to our expression: \[ (2^2)^2 = 2^{2 \times 2} = 2^4 \] ### Step 3: Calculate \( 2^4 \) Now we need to calculate \( 2^4 \): \[ 2^4 = 2 \times 2 \times 2 \times 2 = 16 \] ### Final Answer Thus, the simplified expression \( (2^3 \times 2^{-1})^2 \) evaluates to \( 16 \). ### Conclusion The correct option is **C. \( 16 \)**. ### Explanation of Other Options - **A. \( 4 \)**: This option is incorrect because \( 2^2 = 4 \) is the result of the first part of the expression, but we need to square it to get the final answer. - **B. \( 8 \)**: This option is incorrect because \( 2^3 = 8 \) is the result of \( 2^3 \) alone, but it does not account for the \( 2^{-1} \) or the squaring of the result. - **D. \( 32 \)**: This option is incorrect because \( 2^5 = 32 \) is not relevant to our calculations. The exponentiation was done correctly, leading to \( 2^4 \), not \( 2^5 \). ### Revision Summary - Use the properties of exponents to simplify expressions involving multiplication and powers. - When multiplying like bases, add the exponents. - When raising a power to another power, multiply the exponents. - Always calculate the final result carefully to avoid common pitfalls in exponentiation. By following these steps, you can confidently simplify similar expressions in the future!