Help with rearranging terms

Nov 2008
60
0
Hi everyone!

So I have this equation in my economics book:

1) \(\displaystyle \alpha A K^{\alpha-1} N^{1-\alpha} = r \)

and they simply rearrange the terms to get to this equation:

2) \(\displaystyle K = ( \frac{\alpha A}{r} ) ^{\frac{1}{1-\alpha}} N\)

I don't understand how they get from 1) to 2). I managed to get up to this point, but then I don't know what to do next:

3) \(\displaystyle K^{\alpha - 1} = \frac{r}{\alpha A N ^ {1 - \alpha }} \)

Any help would be greatly appreciated!
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey 22upon7.

You have K^(alpha-1) = (r/alpha*A)*N^(alpha-1) Now taking everything to the power (1/(alpha-1) gives

(K^(alpha-1))^(1/(alpha-1)) = K

= [(r/alpha*A)*N^(alpha-1)]^(1/alpha-1)

= N * (r/alpha*A)^(1/(alpha-1))

= (1/(r/alpha*A))^(1/(1-alpha)) * N since a^(-x) = (1/a)^x and x = alpha - 1 in this case.

= (A*alpha/r)^(1/(1-alpha)) * N
 
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Nov 2008
60
0
Thanks a lot Chiro!

a^(-x) = (1/a)^x and x = alpha - 1 in this case
I didn't know about that, I'll have to try and wrap my head around it - shouldn't be too difficult though! :)