The quantity with units of length only is
Explanation:
We have to combine the following constants:
- , Planck constant, with units
- G, the Newton's gravitational constant, with units
- , the speed of light, with units
The combination of these constant should have units of length only, so with meters (m).
First, we notice that has [kg] in its units, while G has
in its units, so in order to make the [kg] disappear, we have to multiply them and they should have same power, so:
Now we have to make the seconds, [s], disappear. We do that by dividing the new quantity by , so that the new units are:
We are almost done: now the quantity has units of an area, squared meters. Therefore, in order to make it have it units of length, we just take its square root:
Learn more about gravitational constant:
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