To simplify the expression \( (2^3 \times 2^{-5}) \) and express it in standard form, let's break it down step-by-step. ### Step 1: Apply the Laws of Exponents When multiplying numbers with the same base, we can use the law of exponents that states: \[ a^m \times a^n = a^{m+n} \] In our case, the base is \( 2 \), and we have: \[ 2^3 \times 2^{-5} \] Using the law of exponents, we can combine the exponents: \[ 2^{3 + (-5)} = 2^{-2} \] ### Step 2: Convert to Positive Exponent The expression \( 2^{-2} \) can be rewritten using the property of negative exponents, which states: \[ a^{-n} = \frac{1}{a^n} \] Thus, we have: \[ 2^{-2} = \frac{1}{2^2} \] ### Step 3: Calculate \( 2^2 \) Now, we need to calculate \( 2^2 \): \[ 2^2 = 4 \] So, substituting back, we get: \[ 2^{-2} = \frac{1}{4} \] ### Step 4: Express in Standard Form The fraction \( \frac{1}{4} \) can also be expressed as a decimal: \[ \frac{1}{4} = 0.25 \] ### Conclusion Thus, the simplified expression \( (2^3 \times 2^{-5}) \) in standard form is: \[ 0.25 \] ### Answer The correct option is **A. \( 0.25 \)**. ### Explanation of Other Options - **B. \( 0.5 \)**: This option is incorrect because \( 0.5 \) is equivalent to \( \frac{1}{2} \), which does not match our result of \( \frac{1}{4} \). - **C. \( 1 \)**: This option is incorrect because \( 1 \) would imply that the product of the exponents resulted in zero, which is not the case here. - **D. \( 4 \)**: This option is incorrect because \( 4 \) is the result of \( 2^2 \), not \( 2^{-2} \). ### Common Pitfalls - Forgetting to apply the laws of exponents correctly can lead to incorrect simplifications. - Misinterpreting negative exponents can cause confusion; remember that a negative exponent indicates a reciprocal. - Not converting fractions to decimals when required can lead to misinterpretation of the answer format. ### Revision Summary - Use the law of exponents to combine terms with the same base. - Convert negative exponents to positive by taking the reciprocal. - Calculate powers accurately to avoid mistakes. - Express final answers in the required format (standard form).