Rank the results of the following calculations in order of...

Rank the results of the following calculations in order of the number of significant figures, from least to greatest. (a) $6.85 \times 10^{-5} \mathrm{m}+2.7 \times 10^{-7} \mathrm{m}$ (b) $702.35 \mathrm{km}+1897.648 \mathrm{km}$ (c) $5.0 \mathrm{m} \times 4.302 \mathrm{m}$ (d) $(0.040 / \pi) \mathrm{m}$

Rank the results of the following calculations in order of the number of significant figures, from least to greatest.
(a) $6.85 \times 10^{-5} \mathrm{m}+2.7 \times 10^{-7} \mathrm{m}$
(b) $702.35 \mathrm{km}+1897.648 \mathrm{km}$
(c) $5.0 \mathrm{m} \times 4.302 \mathrm{m}$
(d) $(0.040 / \pi) \mathrm{m}$

Key Concepts

Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers in a compact form. It shows a number as a product of a coefficient (between 1 and 10) and a power of ten, while simultaneously maintaining the rules of significant figures. This format is particularly useful in scientific calculations where the magnitude of values can vary widely.

Rounding Rules for Multiplication and Division

When multiplying or dividing numbers, the result should have the same number of significant figures as the factor with the fewest significant figures. This method ensures that the precision of the final answer reflects the precision of the least accurately known factor, avoiding overstatement of the measure's accuracy.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. They include all the non-zero digits, any zeros between them, and any trailing zeros that are specified by a decimal point. Understanding which digits count as significant is critical for accurately reporting measurements and performing calculations where precision is paramount.

Rounding Rules for Addition and Subtraction

When adding or subtracting numbers, the result should be rounded to the least number of decimal places present in any of the measured numbers involved. This rule ensures that the precision of the result does not exceed the precision of the least accurate measurement, which is important for correctly propagating uncertainty through calculations.

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