91Ó°ÊÓ

Understanding the exponent of 10 is crucial when dealing with scientific notation. The term 'exponent of 10' refers to how many times the number 10 is multiplied by itself. For example, in scientific notation, when we say the exponent of 10 is 5, we mean that the decimal point has moved five places. This is represented as the number 10 raised to the power of five, or in mathematical terms, 10 to the 5th power, which is written as \(10^5\).

When converting a large number into scientific notation, the exponent of 10 gives us a concise way to express the magnitude of that number. It's akin to shorthand for counting how many hops the decimal point makes to reduce a number to a figure between 1 and 9.99.

Proper placement of the decimal point is the cornerstone of writing numbers in scientific notation. Essentially, you want to position the decimal point such that the resulting number falls between 1 and 10, while never losing track of the zeros that give the number its size and scale. When dealing with a large number like 987,000, the decimal point begins after the last zero. By moving the decimal point so that it comes after the first non-zero digit, in this case '9', we are restructuring the number into a form that can be easily matched with an exponent of 10.

For 987,000, moving the decimal point right after '9' results in the number 9.87. This is the first step in expressing the original number in the more digestible format of scientific notation.

Significant figures play a significant role in scientific notation, as they represent the precision of a number. These are the digits that carry meaning contributing to its precision. Starting from the first non-zero digit and including all the following digits, they help us define the accuracy of a quantity. In the example of 987,000, the digits 9, 8, and 7 are significant figures because they are the ones that affect the value of this particular number.

When we write numbers in scientific notation, we often consider the significant figures to maintain the original number's precision. So for 987,000, represented as \(9.87 \times 10^5\), the number 9.87 has three significant figures, demonstrating the same precision as the original number.

Writing numbers in scientific notation allows us to concisely represent very large or very small numbers in a standardized form. The process involves a combination of decimal point placement and determining the appropriate exponent of 10. The goal is to write the original number as a product of a number between 1 and 10, and an appropriate power of 10.

For the number 987,000, the steps to write it in scientific notation involved first identifying the significant figures (9.87), then placing the decimal point after the first significant digit to fall within that 1 to 10 range, and finally determining the exponent of 10 by counting how far the decimal has moved from its original position (which was 5 places to the left in this case). Therefore, 987,000 in scientific notation becomes \(9.87 \times 10^5\). This format is not only easier to read but also simplifies mathematical operations.