### 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 is a system of three equations with three unknowns, so I will try to eliminate some variables and solve for $a$.
Since $a^2/b = 1$, I can multiply both sides by $b$ and get $a^2 = b$.
This means I can replace $b$ with $a^2$ in the second equation.
Then I have $b^2/c = 2$ becomes $(a^2)^2/c = 2$, or $a^4/c = 2$.
Multiplying both sides by $c$, I get $a^4 = 2c$.
This means I can replace $c$ with $a^4/2$ in the third equation.
Now I have $c^2/a = 3$ becomes $(a^4/2)^2/a = 3$, or $a^7/4 = 3$.
Multiplying both sides by $4$, I get $a^7 = 12$.
Taking the seventh root of both sides, I get $a = \sqrt[7]{12}$.