Write each of the following in scientific notation:
f. 670 000

Verified step by step guidance

1. Identify the significant figures in the number. For 670,000, the significant figures are 6.7 (ignoring the trailing zeros for now).
2. Determine the power of 10 by counting how many places the decimal point would need to move to the left to place it after the first significant digit (6). In this case, the decimal point moves 5 places to the left.
3. Express the number in the form \( a \times 10^n \), where \( a \) is the significant figures (6.7) and \( n \) is the power of 10 (5).
4. Combine the significant figures and the power of 10 to write the number in scientific notation: \( 6.7 \times 10^5 \).
5. Verify the result by expanding \( 6.7 \times 10^5 \) back into standard notation to ensure it equals 670,000.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small in a compact form. It is written as the product of a number between 1 and 10 and a power of ten. For example, the number 670,000 can be expressed as 6.7 x 10^5, where 6.7 is the coefficient and 5 is the exponent indicating the number of places the decimal point has moved.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. In scientific notation, only the digits in the coefficient are considered significant. For instance, in the number 6.7 x 10^5, both digits '6' and '7' are significant, which helps convey the accuracy of the measurement or value being represented.

Exponents

Exponents are a mathematical notation that indicates how many times a number, known as the base, is multiplied by itself. In scientific notation, the exponent represents the power of ten that scales the coefficient. For example, in 6.7 x 10^5, the exponent '5' means that 10 is multiplied by itself five times, effectively shifting the decimal point five places to the right.