### Instruction: Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$.### Response: This problem involves some algebraic manipulations of fractions and exponents.
One way to approach it is to try to eliminate some of the variables and solve for $a$.
I notice that the first equation gives me a nice expression for $b$ in terms of $a$: $b = a^2$.
If I substitute that into the second equation, I get: $(a^2)^2/c = 2$, or $a^4/c = 2$.
This gives me another expression for $c$ in terms of $a$: $c = a^4/2$.
Now I can substitute both $b$ and $c$ in terms of $a$ into the third equation: $(a^4/2)^2/a = 3$, or $a^7/4 = 3$.
To solve for $a$, I need to isolate it on one side of the equation.
I can do this by multiplying both sides by $4$ and then taking the seventh root: $a = \sqrt[7]{12}$.