To solve the expression \( (2^3 \times 2^{-5})^2 \) and determine which of the given options is equivalent in standard form, we will break down the problem step-by-step. ### Step 1: Simplify the Expression Inside the Parentheses The expression inside the parentheses is \( 2^3 \times 2^{-5} \). When multiplying powers with the same base, we can use the property of exponents that states: \[ a^m \times a^n = a^{m+n} \] Applying this property here: \[ 2^3 \times 2^{-5} = 2^{3 + (-5)} = 2^{3 - 5} = 2^{-2} \] ### Step 2: Raise the Result to the Power of 2 Now we need to raise the result \( 2^{-2} \) to the power of 2. Again, we use the property of exponents that states: \[ (a^m)^n = a^{m \times n} \] So we have: \[ (2^{-2})^2 = 2^{-2 \times 2} = 2^{-4} \] ### Step 3: Identify the Correct Option Now we have simplified the original expression to \( 2^{-4} \). We need to compare this with the options provided: - A. \( 2^{-4} \) - B. \( 2^{-2} \) - C. \( 2^{-1} \) - D. \( 2^{0} \) The correct answer is **A. \( 2^{-4} \)**. ### Step 4: Explain Why Other Options Are Incorrect - **Option B: \( 2^{-2} \)** - This is the result of the first step before raising it to the power of 2. It does not account for the squaring of \( 2^{-2} \), so it is incorrect. - **Option C: \( 2^{-1} \)** - This is not related to our calculations at all. It does not represent any step in the simplification process and is therefore incorrect. - **Option D: \( 2^{0} \)** - This represents the value of 1, which is not related to our expression. The expression \( 2^{-4} \) is not equal to \( 2^{0} \), making this option incorrect. ### Summary of Key Points - We simplified \( 2^3 \times 2^{-5} \) to \( 2^{-2} \) using the property of exponents. - We then raised \( 2^{-2} \) to the power of 2, resulting in \( 2^{-4} \). - The correct answer is **A. \( 2^{-4} \)**, as it matches our final result. - The other options do not correctly represent the simplification process or the final result. ### Revision Summary - Use exponent rules: \( a^m \times a^n = a^{m+n} \) and \( (a^m)^n = a^{m \times n} \). - Simplify expressions step-by-step to avoid mistakes. - Always compare your final result with the provided options carefully. - Remember that negative exponents indicate reciprocals, which can help in understanding the values.