Why do we need to raise the whole pi_3 to power of -1/2?

AI Thread Summary
Raising the whole pi_3 to the power of -1/2 is necessary to maintain dimensional consistency in the formulation of dimensionless terms. This process allows for the creation of new dimensionless combinations from existing parameters while preserving their ratios. The original pi_3 can be altered without losing its fundamental properties, as different exponent combinations can still yield dimensionless results. The discussion emphasizes the importance of understanding how to manipulate these terms in dimensional analysis. Ultimately, the goal is to ensure the terms remain dimensionless while exploring various parameter combinations.
hotjohn
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Homework Statement


in the third photo attached , why do we need to raise the whole pi _3 to power of -1/2 ?
can we do so ? if we do so , the original pi_3 will be changed , right ?

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The Attempt at a Solution

 

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That's impossible to read and maybe a little long. Could you type in a more condensed question in latex?
 
I can't read the attachments either, much too fuzzy.
But maybe I can answer the question. The basic idea is to find dimensionless terms (the 'pi' terms) formed by raising the parameters to various powers and multiplying them together. There is 1 degree if freedom in each such term. E.g. If you came up with m1l1F-1t-2, you could equally write it as m2l2F-2t-4. That combines the same parameters in the same ratios, so is still dimensionless, and represents the same mixture.
Does that explain it?
 
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