round each of the following numbers to 4 sig figs and express the result in scientific notation?

Question

A. 102.53070
B. 656,980
C.0.008543210
D.0.000257870
E.-0.0357202

Answer 1

Signficant digits are numbers that indicate the real value of something. You do not count zeroes that come before or after the numbers that give you the real information.

The number of significant digits tells you about the accuracy of a value.

For example, if someone tells you: The distance from Paris to Krakow (Poland) is roughly 1000 km, you only have one significant digit: the 1. The zeroes are only there to place the 1 in the right place (thousands), they do not give you information. With only 1 significant digit, you get a vague idea of the distance. It could be 1400, 1230, or 862.

The distance form Paris to Krakow is almost 1300 km, you now have two significant digits.
The distance is nearly 1280 km (three digits).
The distance from downtown to downtown: 1275 (4 significant digits).

The distance from Bastille (Paris) to the bridge at Marii Konopnickiej (Krakow) is 1275.7 km (5 digits).

And so on. The more significant digits, the more precise the information.

The number of significant digits indicate the accuracy of a measurement. Sometimes, the zeroes can be part of that, expecially if you make it explicit:
"The distance between these two laser beams is exactly 10.000 cm" (implying that the measurement is less than 10.001 and more than 9.999 cm): that is 5 significant digits, even though most of the digits are zeroes.

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Scientific notation indicates the "order of magnitude" (powers of 10). It is very useful for very large numbers and for unit bases that are based on 10 (like the metric system).

A number would be written as being between 1 and 10 (with the decimal fraction if needed), multiplied by the required power of ten.

For example, the number 1000 is simply 1 x 10^3
The number 250 is written 2.5 x 10^2
The number 0.00351 is 3.51 x 10^-3

Your question involves two steps:
making the number of significant digits 4 (if the 5th number is 5 or greater, you round up the 4th number), and
changing the format to scientific notation.

A. 102.53070
becomes
102.5 (four significant digits)
then
1.025 x 10^2
(hint, you moved the decimal by 2 places to the left)

The order in which you do the two steps does not matter.
Using the steps in the other order (scientific notation first):

D. 0.000257870
becomes
2.5787 x 10^-4
(we moved the decimal 4 places to the right)
then
2.579 x 10^-4
The 5th digit was a 7, therefore the 4th digit was rounded up

Answer 2

You do it.