Thank you for writing. Is there a specific reason why ac does not equal b^2 in this question stem? I plugged it in algebraically and got b^3 on both sides.

Let's answer this question by Process Of Elimination.

I. ac = b²
Must this be true? No.
Consider the case where a = 1, b = 0 and c = 1
This set of values satisfies the given equation (abc = b³),
HOWEVER it is not the case that ac = b²
Plug in the values to get (1)(1) = 0² - NOT TRUE

II. b = 0
Must this be true? No.
Consider the case where a = 1, b = 1 and c = 1
This set of values satisfies the given equation (abc = b³),
HOWEVER it is not the case that b = 0

NOTE: At this point, we need not examine statement III, because none of the answer choices state that only statement III must be true.

However, let's examine statement III for "fun"

III. ac = 1
Must this be true? No.
Consider the case where a = 0, b = 0 and c = 0
This set of values satisfies the given equation (abc = b³),
HOWEVER it is not the case that ac = 1

Answer: A

Short approach

\(abc = b^3\) --> \(b(ac-b^2)=0\) --> EITHER \(b=0\) OR \(ac=b^2\), which means that NONE of the option MUST be true.

For example if \(b=0\) then \(ac\) can equal to any number (not necessarily to 0 or 1), so I and III are not always true, and if \(ac=b^2\) then \(b\) can also equal to any number (not necessarily to 0), so II is not always true.

Answer: A.

Always try to factorize these types of algebra questions instead of cancelling some terms from both sides.
abc=b3 implies b(b2 - ac) = 0 which in turn implies b=0 or b2 = ac but we cannot say either of them must be true.

Case I: ac=b2 cannot fall under "must be true" category because it may so happen that b=0.
Case II: b=0 cannot fall under "must be true" category because it may so happen that ac=b2.
Case III: ac=1 is not even relevant to the context of the question as we cannot deduce ac=1 from any of the given conditions.

Hence, ans is A) None

abc=b^3

I) ac=b^2
If b=0, ac can be whatever value and the equation still holds. Not a must be true.

II) b=0
Lets assume b is not zero. As long as ac=b^2, b can be anything that is not 0. Not a must be true.

III) Same logic in I: If b=0, ac can be whatever value - not necessarily 1. Not a must be true.

A