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?
How does this work?
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Harry_the_cat
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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.
\(\displaystyle k^{2/3}=\frac{1}{3}k\)
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:
\(\displaystyle \frac{k}{k^{\frac{2}{3}}}=3\)
\(\displaystyle k^{1-\frac{2}{3}} =3\)
\(\displaystyle k^{\frac{1}{3}} = 3\)
\(\displaystyle k = 3^3\)
\(\displaystyle k=27\)
Continuing from my correct second line in my post above:
Reciprocating gives:
\(\displaystyle \frac{k}{k^{\frac{2}{3}}}=3\)
\(\displaystyle k^{1-\frac{2}{3}} =3\)
\(\displaystyle k^{\frac{1}{3}} = 3\)
\(\displaystyle k = 3^3\)
\(\displaystyle k=27\)