How does this work?

Adi

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Hey guys pretty simple algebra question. I don't understand how this step works in the solution to an economics problem I'm doing. In particular what operation has been done? Has it been divided by k on both sides? If yes then how does 1/3k/k=3?
 

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MarkFL

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Yes, that's not the result of good algebra. The first equation implies:

\(\displaystyle 3=\frac{k}{k^{2/3}}=k^{1/3}\)
 

Adi

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I'm sorry I still don't understand. Can you explain in a more for dummies way?
 

Harry_the_cat

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Hey guys pretty simple algebra question. I don't understand how this step works in the solution to an economics problem I'm doing. In particular what operation has been done? Has it been divided by k on both sides? If yes then how does 1/3k/k=3?
The second line is incorrect. It should read \(\displaystyle \frac {k^{\frac{2}{3}}}{k} = \frac{1}{3}\) which is equivalent to what MarkFL said
 

MarkFL

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The first equation says:

\(\displaystyle k^{2/3}=\frac{1}{3}k\)

Now, suppose we multiply both sides by 3...we get:

\(\displaystyle 3k^{2/3}=k\)

Now, divide both sides by \(k^{2/3}\)

\(\displaystyle 3=\frac{k}{k^{2/3}}\)

In the second image you posted, they do eventually get there, but they introduce other algebraic errors that magically cancel out the first. That solution is very poorly written.
 

Harry_the_cat

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There are 2 errors in the full "solution" you have supplied which effectively cancel each other out. The first error is from line 1 to 2 and the second is from line 3 to 4.
Continuing from my correct second line in my post above:
Reciprocating gives:
\(\displaystyle \frac{k}{k^{\frac{2}{3}}}=3\)
\(\displaystyle k^{1-\frac{2}{3}} =3\)
\(\displaystyle k^{\frac{1}{3}} = 3\)
\(\displaystyle k = 3^3\)
\(\displaystyle k=27\)
 
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