To simplify the expression \( (3 \times 10^4) \times (2 \times 10^3) \) and express 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: 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 A: \( 6 \times 10^7 \)** - This is actually the correct answer, so it should be noted that the current recorded correct option (B) is incorrect. - **Option B: \( 6 \times 10^8 \)** - This option is incorrect because it suggests that we added the exponents incorrectly. The correct addition of the exponents is \( 4 + 3 = 7 \), not \( 8 \). - **Option C: \( 5.6 \times 10^7 \)** - This option is incorrect because it misrepresents the coefficient. The correct coefficient from our multiplication is 6, not 5.6. - **Option D: \( 5.6 \times 10^8 \)** - This option is incorrect for the same reasons as option B and C. It incorrectly states both the coefficient and the exponent. ### Common Pitfalls - **Misadding Exponents**: A common mistake is to miscalculate the sum of the exponents when multiplying powers of ten. - **Incorrect Coefficient**: Sometimes students might confuse the coefficients, especially if they are not careful with their multiplication. ### 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 exponents to avoid common mistakes.