If $1$cm = $.01$m then shouldn't the square root of $1$cm = the square root of $.01$m but the square root of $1$ = $1$ while the square root of $0.01$ = $0.1$ So my dilemma, is the square root of $1$ centimeter = $0.01$ meters or $0.1$ meters?
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$\begingroup$I think you need to think of the square root of a distance as a new unit. Just as a ${m^2}$ is an area, not a distance. So the following works:
$\sqrt {1m} = \sqrt {100cm} $
$\sqrt m = 10\sqrt {cm} $
$\sqrt {1cm} = \sqrt {cm} $
and
$\sqrt {.01m} = .1\sqrt m = .1\left( {10\sqrt {cm} } \right) = \sqrt {cm} $
So it works out either way.
$\endgroup$ $\begingroup$The square root of $0.01\,\mathrm{m}$ is $0.1\,\mathrm{m}^{1/2}$.
Such quantities whose units have fractional exponents don't have a ready physical interpretation, but can sometimes be useful as intermediates in calculations anyway.
From this calculation we can also see the necessary relation between the units $\mathrm m^{1/2}$ and $\mathrm{cm}^{1/2}$, namely $1\,\mathrm{cm}^{1/2} = 0.1\,\mathrm{m}^{1/2}$.
$\endgroup$ 1 $\begingroup$The side of a square of area $0.01 \text{ m}^2$ is $0.1 \text { m}$. The side of a square of area $1 \text{ cm}^2$ is $1 \text { cm}$. There is no square root of a centimeter-there is (aside from fracals) nothing of dimension $\frac 12$, which is what would be required.
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