# 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\$.

• Good answer. As a person who has taken GRE General Test recently, I just wanted to note that all the variables in GRE General Test are assumed to be real. :) This is explicitly stated in the beginning of the test. If this assumption is lifted, many of the quantitative comparison problems (as above) would have the answer "The relationship cannot be determined from the information given." Just thought this would be beneficial for people taking the exam. – Prism Sep 14 '13 at 19:38
• @Prism I figured as much, but didn't know for sure, so thanks for the information! – Namaste Sep 14 '13 at 19:39
• @amWhy: Nice answer, there is an extra dollar sign at the end. +1 – Amzoti Sep 15 '13 at 14:09