I am trying to work out $(\Pi \circ f)(L)$. The functions are defined:
$ \Pi(y) = -y^4 + 6y^2 - 5 $, and $ f(L) = 5L^\frac23$
I understand the first simplification to:
$-(5L^\frac23)^4 + 6(5L^\frac23)^2 - 5$
But I do not understand the final simplification. Specifically, I do not understand where 625 and 150 come from in the final equation below:
$-625L^\frac83 + 150L^\frac43 - 5$
I can see that $625$ is $5^4 \times 5$, and that $125$ is $5^2 \times 5$, but I don't get why we have 625 and 150 in the equation, rather than just $-5L^\frac83$, for example. Thanks for the help!
You just got a mistake which probably confuses you: $5^4=625$ already. And $5^2=25$, but you still have to multiply with the factor $6$... To take the exponent of a product, you have to multiply the exponents of each factor. You only took the exponent of the unknown variable.