Write the scientific notation for 127.5.
The Correct Answer and Explanation is:
To express the number 127.5 in scientific notation, we need to represent it in the form of ( a \times 10^n ), where ( a ) is a number greater than or equal to 1 and less than 10, and ( n ) is an integer.
Step-by-Step Conversion
- Identify the Decimal Point Movement:
- For the number 127.5, we begin by moving the decimal point to the left until only one non-zero digit remains to its left. In this case, moving the decimal point two places to the left gives us 1.275.
- Count the Number of Places Moved:
- We moved the decimal point 2 places to the left. In scientific notation, moving the decimal to the left indicates a positive exponent.
- Write in Scientific Notation:
- Since we have ( 1.275 ) after moving the decimal point, and we moved it two places to the left, we can express 127.5 in scientific notation as:
[
1.275 \times 10^2
]
Explanation of Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a concise format. It is widely used in fields such as science, engineering, and mathematics because it simplifies calculations and makes it easier to read and compare numbers.
In scientific notation, the number ( a ) represents the significant digits of the original number, while ( 10^n ) shows the scale of the number. For instance, ( 10^2 ) indicates that the decimal point in the number ( a ) should be moved two places to the right to return to the original value.
Using scientific notation helps in various calculations, particularly when dealing with numbers that span multiple orders of magnitude. It allows scientists and mathematicians to avoid writing long strings of zeros, which can lead to errors in communication and calculation. By adhering to the format ( a \times 10^n ), it becomes clear which numbers are significant and how they relate to one another in terms of size and scale.
Thus, the scientific notation for 127.5 is:
[
\boxed{1.275 \times 10^2}
]