To solve the expression \( 2^3 \times 2^4 \) and express it in standard form, we will follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understanding Exponent Rules When multiplying numbers with the same base, we can use the rule of exponents that states: \[ a^m \times a^n = a^{m+n} \] In our case, the base \( a \) is \( 2 \), and the exponents are \( 3 \) and \( 4 \). ### Step 2: Applying the Exponent Rule Using the exponent rule, we can combine the two terms: \[ 2^3 \times 2^4 = 2^{3+4} = 2^7 \] ### Step 3: Expressing in Standard Form Now that we have simplified the expression to \( 2^7 \), we can also convert this to a numerical value if needed. To do this, we calculate \( 2^7 \): \[ 2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \] Calculating this step-by-step: - \( 2 \times 2 = 4 \) - \( 4 \times 2 = 8 \) - \( 8 \times 2 = 16 \) - \( 16 \times 2 = 32 \) - \( 32 \times 2 = 64 \) - \( 64 \times 2 = 128 \) Thus, \( 2^7 = 128 \). ### Step 4: Evaluating the Options Now, let's evaluate the options provided: - **Option A: \( 2^7 \)** - This is correct as we derived \( 2^7 \) from the multiplication. - **Option B: \( 8 \times 16 \)** - This is also equal to \( 128 \) (since \( 8 \times 16 = 128 \)), but it is not in the standard form of \( 2^n \). - **Option C: \( 128 \)** - This is the numerical value of \( 2^7 \), but it is not in the form of \( 2^n \). - **Option D: \( 64 \)** - This is incorrect because \( 64 \) is equal to \( 2^6 \), not \( 2^7 \). ### Conclusion The correct answer is **Option A: \( 2^7 \)** because it directly represents the result of the multiplication in standard form. ### Summary of Key Points - When multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \). - \( 2^3 \times 2^4 = 2^{3+4} = 2^7 \). - The numerical value of \( 2^7 \) is \( 128 \), but the question specifically asks for the standard form. - Other options either represent the same value in a different form or are incorrect. This thorough understanding of exponent rules and careful evaluation of options will help you in similar problems in the future!