To determine which of the given options is expressed in correct scientific notation with the appropriate number of significant figures, we first need to understand the rules of scientific notation and significant figures. ### Understanding Scientific Notation 1. **Definition**: Scientific notation is a way of expressing numbers that are too large or too small in a more manageable form. It is written as \( a \times 10^n \), where: - \( a \) is a number greater than or equal to 1 and less than 10. - \( n \) is an integer (which can be positive or negative). 2. **Significant Figures**: Significant figures are the digits in a number that contribute to its precision. This includes: - All non-zero digits. - Any zeros between significant digits. - Trailing zeros in the decimal portion. ### Analyzing Each Option Now, let's analyze each option to see if it meets the criteria for scientific notation and significant figures. #### Option A: \( 0.00456 \times 10^3 \) - **Scientific Notation Check**: The number \( 0.00456 \) is not in proper scientific notation because it is less than 1. To convert it to scientific notation, we would write it as \( 4.56 \times 10^{-3} \). - **Significant Figures**: The number \( 0.00456 \) has three significant figures (4, 5, and 6). - **Conclusion**: This option is incorrect. #### Option B: \( 3.200 \times 10^2 \) - **Scientific Notation Check**: The number \( 3.200 \) is in proper scientific notation because it is between 1 and 10. - **Significant Figures**: The number \( 3.200 \) has four significant figures (3, 2, 0, and 0). The trailing zeros after the decimal point are significant. - **Conclusion**: This option is correct. #### Option C: \( 5.70 \times 10^1 \) - **Scientific Notation Check**: The number \( 5.70 \) is also in proper scientific notation. - **Significant Figures**: The number \( 5.70 \) has three significant figures (5, 7, and the trailing 0). The trailing zero is significant because it indicates precision. - **Conclusion**: This option is correct, but we need to compare it with others. #### Option D: \( 5000 \times 10^0 \) - **Scientific Notation Check**: The number \( 5000 \) is not in proper scientific notation because it is not expressed as a number between 1 and 10. It should be written as \( 5.0 \times 10^3 \) to be in proper form. - **Significant Figures**: The number \( 5000 \) can have one, two, three, or four significant figures depending on how it is written (e.g., \( 5.0 \) has two significant figures, \( 5.00 \) has three, etc.). Without a decimal point, it is ambiguous. - **Conclusion**: This option is incorrect. ### Summary of Findings - **Correct Options**: B and C are both in correct scientific notation and have the appropriate number of significant figures. - **Best Option**: Between B and C, option B has four significant figures, which is more precise than option C's three significant figures. ### Final Answer **The correct option is B: \( 3.200 \times 10^2 \)**. ### Revision Summary - Scientific notation must have a number between 1 and 10 multiplied by a power of ten. - Significant figures include all non-zero digits, zeros between them, and trailing zeros in decimal numbers. - Options A and D are incorrect due to improper scientific notation, while B is the most precise with four significant figures. - Option C is also correct but less precise than option B.