91Ó°ÊÓ

Significant figures represent the digits in a number that are meaningful in expressing reliable precision, starting from the first non-zero digit. In the example of 678,000: 678 are considered the significant figures. This is because zeros used only to locate the decimal point are not significant.

Understanding significant figures helps us understand the accuracy of a number. These figures clarify which digits carry meaning contributing to the precision of the overall number. When transforming a number to scientific notation, focus on maintaining these significant figures to ensure we preserve the number's original intent and accuracy. Here, we transformed 678,000 into smaller digits with a larger value for clarity and consistency.

A power of ten is a number that is expressed as ten raised to an exponent. In scientific notation, we use powers of ten to express large or small numbers concisely. When converting numbers like 678,000 to scientific notation, the decimal is moved to a position after the first significant digit. In this case, the number becomes 6.78. The decimal point originally at the end of 678,000 moves 5 places to the left.

The power of ten, in this case, is 5, expressed as \(10^5\). This power indicates that the decimal point must move 5 places to the right to return to the original number. Powers of ten allow numbers to be easily compared and understood despite their scale or size.

In transitioning to scientific notation, knowing where to place the decimal point is crucial. The decimal point is moved to create a new number from the significant figures that fall between 1 and 10. For 678,000, the significant figures are 678, so the decimal point is placed right after the first figure, resulting in 6.78. This placement helps create a more manageable and interpretable number.

The placement of the decimal ensures the number is properly adjusted to fit scientific notation format. To understand this effectively, remember the decimal moves based on how many zeros follow the significant figures or non-significant starting digits. This movement determines the power of ten, indicating how many places you moved the decimal. This small adjustment reveals the full magnitude of the original number efficiently, using only the few necessary digits followed by the appropriate power of ten.