Answer: C
A: if ab <0 then:
a<0 and b>0 : then c should be negative to have a^3*b^4*c^7 >0
a>0 and b<0 : then it’s ok because b’s power is even.
So A is possible but not obligatory.

B: it’s possible, But it’s not obligatory.

C: ac is positive: True
Because power of a and c are odd, they both should have same sign, both negative or positibve to have the whole expression positive.

the above statement will be true if any of the following statement are true,

1. a , b , c are all positive numbers

2. if a is +ve, b is -ve and c is +ve

3. if a is -ve, b is +ve and c is -ve

Therefore only option C is always true

I'm confused with this question-answer at this point, why is B not a "must be true" statement?

nvm got it, got tricked

abc: - - + -> +
347: - + + -> -

how come B isn't an answer?

We're told that $a^3b^4c^7>0$
So, it COULD be the case that $a=1$, $b=-1$ and $c=1$

Statement B says $abc$ must be positive
However, if $a=1$, $b=-1$ and $c=1$ then $abc$ is negative

So, we can eliminate B

Cheers,
Brent

if b is equal -1 then it must be positive as rules coz $b^4$. If a or c is equal -1 not the b then we can easily eliminate the answer choice B.

Brent's answer is a really clever one, and a reminder that when you see inequalities like this on the GRE, you should look to manipulate it so that it's still true. You'll likely find a cool shortcut.

The concept being tested here is odd exponents preserve the sign of the base while even exponents do not.

For example

$(-2)^6 =+64$
$(2)^6 = +64$

$(-2)^6 =+64 = (2)^6 = +64$

$(-2)^5 = -32$
$(2)^5 = +32$

$(-2)^5 = -32 \neq (2)^5 = +32$

Now, if all three, $a$,$b$,$c$ are positive, then the inequality is easily satisfied.

But it can also be satisfied if $a$ and $c$ are negative. Explanation below.

Now since $a^3b^4c^7 > 0$, we can conclude that the sign of $b$ is immaterial to satisfying the inequality. However, both $a$ and $c$ have to be negative so that $a^3$ and $c^7$ are negative and when multiplied together become positive to satisfy the inequality.

Therefore, $a$ is negative, $c$ is negative and $b$ can be positive or negative.

A. ab : $a=-ve$, $b=+ve$ or $b=-ve$, thus $ab=+ve$ or $-ve$, thus $ab$ could be negative
B. abc : $a=-ve$, $b=+ve$ or $-ve$,$c=-ve$, thus $abc = +ve$ or $-ve$, thus $abc$ could be positive
C. ac : $a=-ve$ and $c=-ve$, thus $ac=+ve$, thus $ac$ must be positive

Therefore, C is the correct answer.