# When is a radical in its simplest form? Seemingly conflicting answers

Simply put, is a radical in its simplest form if all square factors are removed and radicand is still a (non-square) fraction? Like: $$\frac{1}{42}\sqrt\frac{1}{2}$$

This is the solution given on Khan Academy asking to simplify the below problem: $$-\sqrt\frac{1}{98} + \sqrt\frac{1}{72}$$

However, according to this site, "A radical is in its simplest form when the radicand is not a fraction."

So who's correct?

I tried to remove the fraction anyway. Is my solution correct?

$$\frac{1}{42}\sqrt\frac{1}{2} = \frac{1}{42}\sqrt\frac{1 \times 2}{2 \times 2}$$

$$\frac{1}{42}\sqrt\frac{1 \times 2}{2 \times 2} = \frac{1}{42}\sqrt\frac{2}{4} = \frac{1}{42} \times \frac{1}{2}\sqrt2$$

$$=\frac{1}{84}\sqrt2$$

• The term simplest form is not well-defined: it can mean different things in different circumstances. Your solution is correct for one common definition, though even with that definition some would prefer that you write it $\frac{\sqrt2}{84}$. – Brian M. Scott Sep 18 '16 at 21:15
• How do I judge what form my answer should take? – Shiny_and_Chrome Sep 18 '16 at 21:36
• If you’ve been given an explicit definition, use that. If you’ve not been given a definition, but you have been shown examples, you simply have to use them to make a reasonable guess at what’s wanted. – Brian M. Scott Sep 18 '16 at 21:38
• Assuming that your answer is to be read by a living human being, there is no way of being sure your answer will be accepted: different folks have different tastes. – Lubin Sep 18 '16 at 21:39
• There is not one, single correct answer. However, there are a lot of answers that are wrong if they ask for the simplest form, such as $\frac{2\sqrt2}{168}$. As long as you avoid those, you should be reasonably safe. But again, it depends on whoever is correcting your problems. – Arthur Sep 18 '16 at 21:51