One ordered pair $(a,b)$ satisfies the two equations $ab^4 = 48a^2 b$ and $a^2 b^5 = 6a^3 b$. What is the value of $b$ in this ordered pair? (Note: you may have to use the Tab key to get your cursor into the middle answer box.)
We can rewrite the given equations as $ab^3 = 48a^2 $ and $ab^4 = 6a^3$.
Now, divide the given equations to obtain $b = \frac{48a^2}{6a^3} = \frac{8}{a}$.
Substituting this expression for $b$ into the second equation, we have $\frac{8}{a}a^4 =6a^3 \Rightarrow 8a = 6a^3$.
If $a=0$, then $\frac{8}{a}$ is undefined, so $a\neq0$. Now, we may divide both sides of the equation by $6a$ to obtain $ a^2 = \frac{4}{3}$.
Taking the square root of both sides yields $b=\boxed{\frac{2\sqrt 3}{3}}$ or $b= -\frac{2\sqrt 3}{3}$.