Perform the following calculations, and express the answers with the correct number of significant figures. $$\begin{array}{l}{\text { a. } 2.145+0.002} \\ {\text { b. }(9.8 \times 8.934)+0.0048} \\ {\text { c. }(172.56 / 43.8)-1.825}\end{array}$$

Understanding scientific notation is crucial for handling both very large and very small numbers efficiently. It expresses numbers as a product of two parts: a base number between 1 and 10, and an exponent of 10 written as a power. For example, the number 2,000 can be written in scientific notation as \(2 \times 10^{3}\).

When using scientific notation in calculations, keep in mind that only the base numbers should be directly manipulated while keeping track of the exponents separately. For instance, when you multiply \(2 \times 10^{3}\) by \(3 \times 10^{2}\), you multiply the base numbers (2 and 3) and add the exponents, resulting in \(6 \times 10^{5}\).

Scientific notation not only simplifies the process of dealing with very large or small numbers but also assists in maintaining precision during calculations since it clearly identifies significant figures in the base number.

When dealing with addition and subtraction in scientific measurements, significant figures play a key role in determining the precision of the answer. The rule of thumb is to look at the least precise measurement—meaning the one with the fewest decimal places. This will dictate the number of decimal places in your final answer.

For example, consider the sum of the numbers 6.341 and 0.12. Since 0.12 has only two decimal places, our result should also be rounded to two decimal places, leading to 6.46. This ensures that the answer is not represented with a false precision that was not present in the initial data. Always align the decimal places before performing addition or subtraction and round off the final answer to the appropriate number of decimal places to respect the significant figure rules.

During multiplication and division, the principle of significant figures differs from that of addition and subtraction. Instead of aligning decimal places, you must count the number of significant figures in each term you're multiplying or dividing. The term with the least number of significant figures determines the number of significant figures in your final answer.

As an example, when multiplying 4.56 (three significant figures) by 0.92 (two significant figures), the answer will be rounded to two significant figures, resulting in 4.2. Similarly, if dividing 15.0 (three significant figures) by 3.221 (four significant figures), the quotient would be rounded to three significant figures. It is essential to ensure that the final answer reflects the inherent precision of the least precise term involved in the calculation.