Help needed with exponents

[j4m]

I'm trying to understand how L is solved in this -- how -z/pi*alpha^alpha-1 (lambda is in the pic but is equal to 1 so I ignored it here) is flipped so that it equals pi*alpha^(1/1-alpha)/z. This result was taken from the answers but I'd appreciate if someone could explain the process/laws that cause this.

For context, this is from using partial derivatives to find first order conditions in an economic problem.

Cheers.

 

[Dr.Peterson]

I'm trying to understand how L is solved in this -- how -z/pi*alpha^alpha-1 (lambda is in the pic but is equal to 1 so I ignored it here) is flipped so that it equals pi*alpha^(1/1-alpha)/z. This result was taken from the answers but I'd appreciate if someone could explain the process/laws that cause this.

For context, this is from using partial derivatives to find first order conditions in an economic problem.

Cheers.

View attachment 32910

Are you saying this is a copy of someone's work that you are trying to understand? Or is the boxed equation the supposedly correct answer, and the rest is your own work (which could perhaps be wrong)?

It's clear that the two expressions for L are not equivalent.

Can you show us the actual problem, and any work or answers provided to you, in print (your handwriting is in an unfamiliar style, and therefore hard for me to read), and tell us more about what you have written here? Everything is rather cryptic.

 

[j4m]

Are you saying this is a copy of someone's work that you are trying to understand? Or is the boxed equation the supposedly correct answer, and the rest is your own work (which could perhaps be wrong)?

It's clear that the two expressions for L are not equivalent.

Can you show us the actual problem, and any work or answers provided to you, in print (your handwriting is in an unfamiliar style, and therefore hard for me to read), and tell us more about what you have written here? Everything is rather cryptic.

Sure, thanks for the help. I'll attach what we've been given but the main part is the second constraint which is used to find the first order conditions.





I'm struggling to follow how L is solved in the answer

 

[Steven G]

How can 1 + λ and 1 - λ both equal 0???

 

[j4m]

How can 1 + λ and 1 - λ both equal 0???

1b is the main part. The bit before uses the answer sheet posted above (for 1 = lambda) and my failed attempt to solve it (-1 = lambda).

 

[Dr.Peterson]

Sure, thanks for the help. I'll attach what we've been given but the main part is the second constraint which is used to find the first order conditions.

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View attachment 32919

I'm struggling to follow how L is solved in the answer

Does this mean you are asking about this step?

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If so, they just divided by [imath]\alpha[/imath], and raised both sides to the -1 power (changing the exponent to [imath]1-\alpha[/imath] and taking the reciprocal of the RHS) and raised both sides to the reciprocal power to isolate l.