Perform the following calculations and report each answer with the correct number of significant figures. $$(a) 628 \times 342$$ $$\text { (b) }\left(5.63 \times 10^{2}\right) \times\left(7.4 \times 10^{3}\right)$$ $$\text { (c) } \frac{28.0}{13.483}$$ $$\text { (d) } 8119 \times 0.000023$$ $$\text { (e) } 14.98+27,340+84.7593$$ $$(f) 42.7+0.259$$

When you multiply numbers, like the ones in exercise (a), you must keep in mind the principles of significant figures. This rule states that your result can only be as precise as your least precise factor. For example, when multiplying 628 with 342, since both numbers are expressed with three significant figures, your final answer should also have three significant figures. This ensures that your answer does not present a false sense of precision.

Remember that when more digits appear in the product than are allowed by these rules, you need to round the result to maintain the appropriate number of significant figures. This contributes to the accuracy and reliability of scientific measurements and calculations.

Exponential multiplication, as shown in exercise (b), can initially seem daunting, but it essentially follows the same rules as regular multiplication. Start by multiplying the coefficients (5.63 and 7.4 in this case). Next, manage the exponents separately—by adding them together. It's crucial to keep the least number of significant figures from the coefficients, in this scenario, only two. This is because the final result's precision in scientific notation should reflect the precision of the less precise number you started with.

Dividing numbers, as in exercise (c), also requires attention to significant figures. When you divide 28.0 by 13.483, even though the latter has more significant figures, the quotient ought to be rounded off to the lowest number of significant figures of the two, which is three in this case. This practice prevents overrepresentation of the precision in the result. Rounding, which is an intrinsic part of handling significant figures, plays a vital role here to ensure the quotient does not appear more accurate than the initial measurements.

Unlike multiplication or division, addition and subtraction focus on the number of decimal places rather than significant figures. Take exercise (e) as an example. When summing up 14.98, 27340, and 84.7593, line up the decimal points and the answer should be reported with two decimal places, which is the least number of decimal places (look at 14.98).

Addition with significant figures adheres to the principle that the final answer should not be more precise than the least precise term in the sum. Exercise (f) further illustrates this point, demonstrating this technique with one decimal place as shown by the number 42.7.

When rounding numbers to the correct number of significant figures, it's vital to understand and apply the proper rounding rules. If the digit to be dropped (the one immediately to the right of the last significant figure) is less than 5, simply drop it; if it's 5 or greater, round up the last significant figure. These rounding rules apply to all operations, be it multiplication or addition, ensuring the results properly reflect the significant figure constraints of the data. Correctly rounding maintains the integrity of the calculated results and communicates the associated precision accurately.