Express the product of 0.00043 and 2000 in standard form.

Analysis

To solve this problem, we need to find the product of two numbers given in decimal and whole number forms, and then convert the final result into standard form (also known as scientific notation). Standard form is written as $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

Concept

1. Multiplication: Multiply the two numbers directly.
2. Standard Form Conversion: Move the decimal point so that only one non-zero digit remains to the left of the decimal point. Count how many places the decimal point moved to determine the power of 10.

Step-by-Step Solution

Step 1: Multiply the numbers We are calculating $0.00043 \times 2000$.

To make this easier, we can rewrite the numbers: $0.00043 = 43 \times 10^{-5}$ $2000 = 2 \times 10^3$

Now, multiply them: $(43 \times 10^{-5}) \times (2 \times 10^3)$ $= (43 \times 2) \times (10^{-5} \times 10^3)$ $= 86 \times 10^{-5 + 3}$ $= 86 \times 10^{-2}$

Step 2: Convert to standard form The current result is $86 \times 10^{-2}$. In standard form, the number must be between 1 and 10. We need to change $86$ to $8.6$. To change $86$ to $8.6$, we divide by 10, which means we must multiply the power of 10 by 10 (or add 1 to the exponent):

$86 \times 10^{-2} = (8.6 \times 10^1) \times 10^{-2}$ $= 8.6 \times 10^{1 + (-2)}$ $= 8.6 \times 10^{-1}$

Conclusion

Comparing our result ($8.6 \times 10^{-1}$) with the given options, it matches option C.

Exam Tip

When multiplying numbers with decimals and zeros, convert them to powers of 10 first. It prevents errors caused by miscounting decimal places. Remember: if you move the decimal point to the left, the exponent increases; if you move it to the right, the exponent decreases.