# How to compare $(\sqrt{6a})(5\sqrt{3})$ to $\sqrt{200a}$?

This is a gre question: I have chosen the answer A because the quantity A can be written as : $\sqrt{6 \times 25 \times 3 a}$ which is $\sqrt{450 a}$ always greater that the option Quantity B. But someone clams that the relationship can not be given from the above statement because difference values of a gives different answer. In the Gre question are we allowed to take the value 0 and any other values?

• Well, you don't know whether $a > 0$ or $a = 0$, or maybe $a$ is even complex, so you can't say from the information given. – Daniel Fischer Sep 14 '13 at 19:27
• I hope this was on a practice exam and not the real thing because otherwise this would be very unethical. – Cameron Williams Sep 14 '13 at 19:28
• You can take any value of a you wish, since a is not defined, 0 or -ve reals or what ever else you wish.:) – Ram Sep 14 '13 at 19:36

Yes, $a$ can take on the value of $0$ and/or any other (assuming positive real) value. $a$ is an unknown, and as such, while you're answer is correct for a non-zero (assuming positive) value $a$, option $(c)$ is correct if $a = 0$.
Hence, option $(d)$ is the correct answer.
Of course, if $a$ is complex, (and non-real) then we have no way of comparing quantity $A$ or quantity B\$.