To solve the expression \( 2^{3} \times 2^{-5} \) and express it as a single power of 2, we can use the properties of exponents. Let's break this down step-by-step. ### Step 1: Understanding the Properties of Exponents One of the key properties of exponents states that when you multiply two powers with the same base, you can add their exponents. This property can be written as: \[ a^m \times a^n = a^{m+n} \] In our case, the base \( a \) is 2, and we have: - \( m = 3 \) (from \( 2^3 \)) - \( n = -5 \) (from \( 2^{-5} \)) ### Step 2: Applying the Property Now, we can apply the property of exponents to our expression: \[ 2^{3} \times 2^{-5} = 2^{3 + (-5)} \] ### Step 3: Performing the Addition Next, we need to perform the addition of the exponents: \[ 3 + (-5) = 3 - 5 = -2 \] ### Step 4: Writing the Final Answer Now that we have simplified the exponent, we can write the expression as: \[ 2^{3} \times 2^{-5} = 2^{-2} \] ### Conclusion Thus, the value of \( 2^{3} \times 2^{-5} \) expressed as a single power of 2 is: \[ \boxed{2^{-2}} \] ### Explanation of the Options Now, let's analyze the options provided: - **A. \( 2^{-2} \)**: This is the correct answer, as we derived it from the properties of exponents. - **B. \( 2^{2} \)**: This option is incorrect. It suggests that the result of the multiplication is a positive exponent, which is not the case here. The calculation clearly shows that the exponent is negative. - **C. \( 2^{3} \)**: This option is also incorrect. It represents the original term \( 2^{3} \) without considering the effect of multiplying by \( 2^{-5} \). The multiplication changes the exponent, and thus this option does not reflect the correct result. - **D. \( 2^{-3} \)**: This option is incorrect as well. It suggests a different exponent than what we calculated. The correct exponent after performing the addition of \( 3 \) and \( -5 \) is \( -2 \), not \( -3 \). ### Common Pitfalls - **Forgetting to add exponents**: A common mistake is to forget that when multiplying powers with the same base, you need to add the exponents. - **Misinterpreting negative exponents**: Remember that a negative exponent indicates a reciprocal. For example, \( 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \), but in this case, we are simply expressing the result as a power of 2. ### Revision Summary - Use the property \( a^m \times a^n = a^{m+n} \) for multiplying powers with the same base. - Carefully add the exponents when applying this property. - Negative exponents indicate a reciprocal, but in this case, we are simply expressing the result as a power of 2. - The correct answer for \( 2^{3} \times 2^{-5} \) is \( 2^{-2} \).