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I'm stuck in a really stupid question. It seems like the meaning of "centi" is chancing meaning. Because if:

$ r = 5 cm = 5 \cdot 10^{-2} m $

so $c = 10^{-2}$

But say i need to take the cubic root of r:

$r^3 = 125 cm^3$, this is true. But in my head it should be:

$r^3 = (5cm)^3 = 5^3 c^3m^3 = 125\cdot10^{-6} m^3 $

So while the true answer is centi meters^3, the calculation only makes sencse if the answer is in micro meters^3...

Would love your inupt i am going nuts

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    $\begingroup$ I don't understand your last line. Everything looks fine to me? 125 cm^3 = 125*10^(-6) m^3 as stated correctly. $\endgroup$
    – Noldig
    Commented Feb 21, 2019 at 18:27
  • $\begingroup$ Do you understand that if there are $3$ feet in a yard, there are $9$ square feet in a square yard? $\endgroup$
    – knzhou
    Commented Feb 21, 2019 at 18:43
  • $\begingroup$ You are getting micro (cubic meters). you are not getting (micrometers) cubed (which would be 10 to the -18). $\endgroup$
    – ohwilleke
    Commented Feb 21, 2019 at 18:45
  • $\begingroup$ @Noldig last line is what i think should be right. knzhou thank you, but i understand now that SI prefixing are combining, like what indiaria wrote $\endgroup$
    – nammerkage
    Commented Feb 21, 2019 at 18:51
  • $\begingroup$ a centi (cubic meter) is not the same as a cubic (centi meter). $\endgroup$
    – John Alexiou
    Commented Feb 21, 2019 at 21:42

2 Answers 2

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SI prefixes are combining, so when you say $\text{cm}^3$, you're actually saying $(\text{cm})^3 = (10^{-2} \text{m})^3 = 10^{-6} \text{m}^3$, similar to how $\mathrm{d}^2y/\mathrm{d}x^2$ refers to $\mathrm{d}^2y/(\mathrm{d}x)^2$ and not $\mathrm{d}^2y/\mathrm{d}(x^2)$. It might help if you if you think it as a different unit "$\text{cm}$" rather than "$\text{c}\cdot\text{m}$". If in doubt you can always expand and factorize it:

$$ 140 \text{km}^2 = 140\times(10^3 \text{m})^2 = 1.4\times 10^8 \text{m} \\ = 1.4\times 10^{8}\times (10^{6})^2\times(10^{-6}\text{m})^2 \\ = 1.4\times 10^{8+6\times 2}(10^{-6}\text{m})^2 = 1.4\times 10^{20} \text{µm}^2 $$

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A cubic micron is $(10^{-6})^3 = 10^{-18}$ cubic meters. Your mistake was saying a micro-cubic-meter was a cubic micrometer (cubic micron).

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