To simplify the expression \( (3 \times 10^4) \times (2 \times 10^3) \) and express the answer in standard form, let's break it down step-by-step. ### Step 1: Understand the Components The expression consists of two parts: - \( 3 \times 10^4 \): This means 3 multiplied by \( 10^4 \) (which is 10,000). - \( 2 \times 10^3 \): This means 2 multiplied by \( 10^3 \) (which is 1,000). ### Step 2: Multiply the Coefficients First, we multiply the numerical coefficients (the numbers in front): \[ 3 \times 2 = 6 \] ### Step 3: Multiply the Powers of Ten Next, we multiply the powers of ten. When multiplying powers of ten, we add the exponents: \[ 10^4 \times 10^3 = 10^{4+3} = 10^7 \] ### Step 4: Combine the Results Now, we combine the results from Steps 2 and 3: \[ (3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7 \] ### Step 5: Express in Standard Form The result \( 6 \times 10^7 \) is already in standard form, which is a number between 1 and 10 multiplied by a power of ten. ### Final Answer Thus, the simplified expression is: \[ \boxed{6 \times 10^7} \] ### Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect: - **Option B: \( 6 \times 10^8 \)**: This option suggests that we added the exponents incorrectly (4 + 3 = 8 instead of 7). This is a common mistake when multiplying powers of ten. - **Option C: \( 6 \times 10^6 \)**: This option indicates that the multiplication of the powers of ten was done incorrectly, possibly subtracting instead of adding the exponents (4 - 3 = 1, which is incorrect). - **Option D: \( 5.5 \times 10^7 \)**: This option suggests a miscalculation of the coefficient. The correct multiplication of the coefficients is \( 3 \times 2 = 6\), not 5.5. ### Common Pitfalls - **Forgetting to add exponents**: When multiplying powers of ten, always remember to add the exponents. - **Miscalculating coefficients**: Ensure that you multiply the numerical parts correctly. - **Not expressing in standard form**: Always check if your final answer is in the correct format (a number between 1 and 10 multiplied by a power of ten). ### Revision Summary - To multiply numbers in scientific notation, multiply the coefficients and add the exponents of the powers of ten. - The expression \( (3 \times 10^4) \times (2 \times 10^3) \) simplifies to \( 6 \times 10^7 \). - Always check that your final answer is in standard form. - Be cautious of common mistakes, especially with exponent rules and coefficient calculations.