Indices help

How do I get from:

a(C2/C1)^1-a/(1-a)(C1/C2)^a

to: a/1-a C2/C1

help

So you have:

$\displaystyle\frac{a\left(\frac{C2}{C1}\right)^{1-a}}{(1-a)\left(\frac{C1}{C2}\right)^a}$

Multiply top and bottom by (C2/C1)^a, i.e. the inverse of the 2nd fraction in the denominator.

and the rest should be trivial.

I don't normally do fully worked solutions, but I think that might be the best thing for you in this case.

You do need to familariise yourself with the rules for manipulating indices. Write them down separately and make sure you understand them all.

From $\displaystyle\frac{a\left(\frac{C2}{C1}\right)^{1-a}}{(1-a)\left(\frac{C1}{C2}\right)^a}$

Multiplying top and bottom by (C2/C1)^a.

$\displaystyle\frac{a\left(\frac{C2}{C1}\right)^{1-a}\left(\frac{C2}{C1}\right)^a}{(1-a)\left(\frac{C1}{C2}\right)^a \left(\frac{C2}{C1}\right)^a}$

We can then add the indices on the top, since what's in the two brackets is the same.

And we can combine the two brackets on the right in the denominator as they are to the same power.


$\displaystyle\frac{a\left(\frac{C2}{C1}\right)^{1-a+a}}{(1-a)\left(\frac{C1}{C2}\times\frac{C2}{C1}\right)^a}$

We then simplify:



And that gives

$\displaystyle\frac{a}{1-a}\frac{C2}{C1}$