Let dim dim ²= ³ ~ ²> ³. Prove that = ° ²= ³ š > ° ²> ³ rad rad implies = š>

We start by expressing dim dim ²= ³ in terms of logarithms:

log(dim²) = log(³)

2log(dim) = log(³)

log(dim) = log(³)/2

Similarly, we can express = ° ²= ³ š > ° ²> ³ in terms of logarithms:

log() = 2log(°)

log() = 3log(°)

log(š) > 2log(°)

log(š) > 3log(°)

Now, we want to prove that = š> . Again, we use logarithms:

log() = 2log(°)

log(š) = 3log(°)

Dividing the second equation by the first, we get:

log(š)/log() = 3/2

Taking the square root of both sides, we get:

log(š)/log()^(1/2) = √3

Now, we use the fact that log(dim) = log(³)/2:

log(³)/2log()^(1/2) = √3

Squaring both sides, we get:

log(³)/4log() = 3

Multiplying both sides by 4log(), we get:

log(³) = 12log()

Dividing both sides by log(), we get:

log(³)/log() = 12

Which means:

= ³

Therefore, we have proven that = š> , given dim dim ²= ³ ~ ²> ³ and = ° ²= ³ š > ° ²> ³ rad rad.