There are a total of 463,100 books in a library. What is the value of the exponent when this number is written in scientific notation?
The Correct Answer and Explanation is :
To express the number 463,100 in scientific notation, we need to rewrite it in the form of ( a \times 10^n ), where ( a ) is a number between 1 and 10, and ( n ) is an integer representing the exponent of 10.
Step 1: Identify the significant digits.
The significant figures in 463,100 are 4, 6, and 3. To convert this number into scientific notation, we need to move the decimal point so that only one non-zero digit appears to the left of it.
Step 2: Move the decimal point.
If we write 463,100 as 4.63100, we see that we have moved the decimal point 5 places to the left (from the end of the number). This gives us the base number ( a = 4.63100 ) (which we can round to 4.63 for simplicity).
Step 3: Determine the exponent.
Since we moved the decimal point 5 places to the left, the exponent ( n ) will be +5. This is because moving the decimal left signifies a positive exponent in scientific notation.
Step 4: Write the final expression.
Combining these components, we express 463,100 in scientific notation as:
[
4.63100 \times 10^5
]
Conclusion
The value of the exponent ( n ) when 463,100 is written in scientific notation is 5. Scientific notation is particularly useful in mathematics and science as it allows for easier reading and calculation with large numbers. It simplifies multiplication and division, as you can multiply or divide the coefficients and add or subtract the exponents, making complex calculations more manageable. This notation is also helpful for accurately representing very small or very large numbers in a standardized form.