The correct option is **A. \( 3^6 \)**. ### Step-by-Step Explanation 1. **Understanding Exponents**: - In mathematics, when we multiply two numbers with the same base, we can add their exponents. This is a fundamental property of exponents. The general rule is: \[ a^m \times a^n = a^{m+n} \] - Here, \( a \) is the base, and \( m \) and \( n \) are the exponents. 2. **Applying the Rule**: - In the expression \( 3^4 \times 3^2 \), both terms have the same base, which is 3. - We can apply the exponent rule: \[ 3^4 \times 3^2 = 3^{4+2} \] - Now, we simply add the exponents: \[ 4 + 2 = 6 \] - Therefore, we have: \[ 3^4 \times 3^2 = 3^6 \] 3. **Final Answer**: - The expression \( 3^4 \times 3^2 \) simplifies to \( 3^6 \), which corresponds to option **A**. ### Why Other Options Are Incorrect - **Option B: \( 3^8 \)**: - This option suggests that we would add the exponents as \( 4 + 4 \), which is incorrect. The correct addition is \( 4 + 2 \), leading to \( 3^6 \), not \( 3^8 \). - **Option C: \( 3^{12} \)**: - This option implies that we would add the exponents as \( 4 + 8 \) or some other incorrect combination. The correct addition is \( 4 + 2 \), which gives \( 3^6 \), not \( 3^{12} \). - **Option D: \( 3^2 \)**: - This option suggests that we would somehow reduce the exponent, which is not how exponent multiplication works. The correct operation leads to an increase in the exponent, resulting in \( 3^6 \), not \( 3^2 \). ### Common Pitfalls - **Forgetting the Rule**: A common mistake is forgetting the property of exponents when multiplying. Always remember to add the exponents when the bases are the same. - **Misreading the Problem**: Ensure that you correctly identify the base and the exponents involved in the multiplication. - **Calculation Errors**: Double-check your arithmetic when adding exponents to avoid simple mistakes. ### Revision Summary - When multiplying powers with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \). - For \( 3^4 \times 3^2 \), add the exponents: \( 4 + 2 = 6 \). - The correct equivalent expression is \( 3^6 \). - Always verify your understanding of exponent rules to avoid common mistakes.