For example when we look at cm^3: the multiplier is then (10^-2)^3 so why don't we write (cm)^3 instead?

  • $\begingroup$ what's the difference between $cm^3$ and $(cm)^3$? $\endgroup$ – Señor O Oct 12 '17 at 20:00
  • $\begingroup$ Are you interpreting "cm" as a product of two algebraic values and interpreting $cm^3$ as $c×(m^3)$? Units of physical properties are not usually interpreted this way, the "cm" (or other unit) is a single symbol on which any power acts in its entirety. $\endgroup$ – IanF1 Oct 12 '17 at 20:00
  • $\begingroup$ Duplicate: physics.stackexchange.com/questions/128627/…? ::blinks:: I recalled that we'd had this question before, but I didn't remember answering it. $\endgroup$ – dmckee --- ex-moderator kitten Oct 12 '17 at 21:41

The main reason is notational simplicity, to avoid the usage of an excessive number of parenthesis. To do this, we define a conventional "order of precedence" between prefix symbols and units. As you do with mathematical expressions, you can then reinstate parenthesis following this conventional order of precedence.

The above rules are detailed in the SI brochure, §3.1. The relevant part is the following (emphasis mine):

The grouping formed by a prefix symbol attached to a unit symbol constitutes a new inseparable unit symbol (forming a multiple or submultiple of the unit concerned) that can be raised to a positive or negative power and that can be combined with other unit symbols to form compound unit symbols.

This leads to the interpretation

$$1\,\mathrm{cm^3} = 1\,(\mathrm{cm})^3.$$

Of course, this is not the only possible choice, and we could have chosen to have the prefix multiplying the unit and the exponent, but, in this case, consider the interpretation of the following expression (in red, to highlight that it's not the one in use):

$$\color{red}{\frac{1\,\mathrm{cm^3}}{1\,\mathrm{cm^2}} = \frac{10^{-2}\,\mathrm{m^3}}{10^{-2}\,\mathrm{m^2}} = 1\,\mathrm{m}}.$$

Isn't it a tad confusing?

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  • $\begingroup$ One little nitpick: I think your first equation should technically be $1\ \mathrm{cm}^3 = 1\ (\mathrm{cm})^3$. $\endgroup$ – David Z Oct 12 '17 at 20:27
  • $\begingroup$ @MassimoOrtolano, 1 cubic centimeter is NOT equal to 1/100 cubic meters. It's equal to 1/1,000,000 of a cubic meter. The same applies to the number of square centimeters in 1 square meter (e.g., 1 $m^2$ = 10,000 $cm^2$) $\endgroup$ – David White Oct 12 '17 at 21:05
  • $\begingroup$ @DavidWhite In fact, the last example shows what would happen with such an interpretation ;-) $\endgroup$ – Massimo Ortolano Oct 12 '17 at 21:17

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