To simplify the expression \( (2^3 \times 2^4) \div 2^5 \) using the laws of indices, let's break it down step by step. ### Step 1: Apply the Law of Indices for Multiplication The first part of the expression is \( 2^3 \times 2^4 \). According to the laws of indices, when you multiply two powers with the same base, you add the exponents. This can be expressed as: \[ a^m \times a^n = a^{m+n} \] In our case: \[ 2^3 \times 2^4 = 2^{3+4} = 2^7 \] ### Step 2: Rewrite the Expression Now, we can rewrite the original expression using the result from Step 1: \[ (2^3 \times 2^4) \div 2^5 = 2^7 \div 2^5 \] ### Step 3: Apply the Law of Indices for Division Next, we need to simplify \( 2^7 \div 2^5 \). According to the laws of indices, when you divide two powers with the same base, you subtract the exponents. This can be expressed as: \[ \frac{a^m}{a^n} = a^{m-n} \] Applying this to our expression: \[ 2^7 \div 2^5 = 2^{7-5} = 2^2 \] ### Final Result Thus, the simplified result of the expression \( (2^3 \times 2^4) \div 2^5 \) is: \[ 2^2 \] ### Conclusion The correct option is **A. \( 2^2 \)**. ### Explanation of Other Options - **B. \( 2^0 \)**: This option is incorrect because \( 2^0 = 1 \), which does not match our result of \( 2^2 \). - **C. \( 2^1 \)**: This option is also incorrect. \( 2^1 = 2 \), which is not the result we obtained. - **D. \( 2^5 \)**: This option is incorrect as well. \( 2^5 = 32 \), which is not related to our simplified expression. ### Common Pitfalls - **Forgetting to add or subtract exponents**: It's crucial to remember the correct operations (addition for multiplication and subtraction for division). - **Misapplying the laws of indices**: Ensure that the base remains the same when applying these laws. ### Revision Summary - Use \( a^m \times a^n = a^{m+n} \) for multiplication of powers. - Use \( \frac{a^m}{a^n} = a^{m-n} \) for division of powers. - Always check that the base is the same when applying these laws. - The final answer for \( (2^3 \times 2^4) \div 2^5 \) is \( 2^2 \).