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So the volume of a cube is V = l x w x h. So a cube with sides 2 is 2 x 2 x 2 = 8

Ok, so now we have a cube with sides 1/4. So 1/4 x 1/4 x 1/4 = 1/64. Why is the volume smaller than the side lengths in this case?8 > 2 but 1/64 < 1/4

Put another way, let's say the cube has side lengths of 1/4 foot. That is the same as 3 inches. If you calculate the volume of the cube in inches, you get 27 inches cubed. But if you calculate it in feet you get 1/64 feet cubed. 27 inches > 1/64 foot. Why aren't they equal?

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3 Answers

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Consider a cube with a side length of $1m$. Its volume is $1m^3$. Written in centimetres, its side length is $100cm$, and as such, its volume is $1000000cm^3$. But $1\neq 1000000$. How can this be so?

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You cannot compare generalized volumes in spaces of different dimension. It makes no sense whatsoever, for instance, to say that an area of $1\ m^2$ is "larger" than a volume of $.00000001\ m^3$.

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Suppose $a > 0$.

When is $a^2 > a$? Well, since $a$ is positive, you can divide by $a$ both sides of the inequality and get $a > 1$. And conversely for positive $a$, if $a > 1$ then you can multiply both sides by $a$ and get $a^2 > a$. So the square of a number is greater than the number if and only if the number is greater than 1. Cubes and higher powers work the same way.

When is $a^2 < a$? Again, since $a$ is positive, you can divide both sides by $a$ and get $a < 1$. And conversely for positive $a$, if $a < 1$ then you can multiply both sides by $a$ and get $a^2 < a$. So the square of a number is less than the number if and only if the number is less than 1. Cubes and higher power work the same way.

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