To simplify the expression \( (3 \times 10^4) \times (2 \times 10^3) \) and write it 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: Apply the Associative Property of Multiplication We can rearrange and group the terms in the expression: \[ (3 \times 10^4) \times (2 \times 10^3) = (3 \times 2) \times (10^4 \times 10^3) \] ### Step 3: Multiply the Coefficients Now, multiply the coefficients (the numbers in front): \[ 3 \times 2 = 6 \] ### Step 4: Multiply the Powers of Ten Next, we apply the property of exponents that states \( a^m \times a^n = a^{m+n} \): \[ 10^4 \times 10^3 = 10^{4+3} = 10^7 \] ### Step 5: Combine the Results Now, we can combine the results from Steps 3 and 4: \[ (3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7 \] ### Step 6: Write 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. ### Conclusion: Identify the Correct Option The final simplified expression is \( 6 \times 10^7 \). Therefore, the correct option is: **A. \( 6 \times 10^7 \)** ### Explanation of Incorrect Options - **B. \( 6 \times 10^6 \)**: This option is incorrect because it suggests that the power of ten is 6 instead of 7. This would imply a much smaller number (6,000,000) than what we calculated. - **C. \( 5.4 \times 10^7 \)**: This option is incorrect because it uses a coefficient of 5.4 instead of 6. The multiplication of the coefficients was done correctly, yielding 6. - **D. \( 5.4 \times 10^6 \)**: This option is incorrect for the same reasons as C; it has both the wrong coefficient and the wrong power of ten. ### Common Pitfalls - **Forgetting to add the exponents**: When multiplying powers of ten, always remember to add the exponents. - **Miscalculating the coefficients**: Ensure that you multiply the coefficients correctly; small errors can lead to incorrect answers. ### Revision Summary - To multiply numbers in scientific notation, multiply the coefficients and add the exponents. - The result should be expressed in standard form, which is a number between 1 and 10 multiplied by a power of ten. - Always double-check your calculations for both the coefficients and the powers of ten. - Familiarize yourself with the properties of exponents to avoid common mistakes.