+239 1) Fully simplify \(\sqrt{49-20\sqrt{6}}\)
2) Fully simplify \(\sqrt {14 + 8\sqrt {3}}\)
+118687 \(\sqrt{49-20\sqrt6}\\ =\sqrt{25+24-20\sqrt6}\\ \text{This might be helpful because 25 is a square and 24 is a multiple of 6}\\ =\sqrt{25-20\sqrt6+4*6}\\ =\sqrt{5^2-20\sqrt6+(2\sqrt6)^2}\\ =\sqrt{5^2-2*5*2\sqrt6+(2\sqrt6)^2}\\ =\sqrt{(5-2\sqrt6)^2}\\ =|5-2\sqrt6|\\ =5-2\sqrt6\\\)
Now maybe you can attempt the second one.
Start by factoring out the 2.
+4622 1. I'll just try this problem...I don't know if I'll get it !
Let \(\sqrt{49-20\sqrt{6}}=a-b\).
We can square both sides to get, \(49-20\sqrt{6}=(a-b)^2\).
Expanding this, gives us \(49-20\sqrt{6}=a^2+b^2-2ab\).
Now, by a bit of matching, we can see that \(a^2-b^2=49\) and \(-2ab=-20\sqrt{6}\) (Same thing as \(2ab=20\sqrt{6}\))
This might be a bit of tedious work, but by a bit of inspection \(a=5\) and \(b=2\sqrt{6}\) work.
Thus, the answer is \(a-b=\boxed{5-2\sqrt{6}}.\)
+118687 Hi Tertre,
I've only learnt this recently and I do not do it quite like that. My technique is pretty bad.
But your way looks really good.
Most times I do not think it would work. I mean i doubt that questions like this can be simplified all that often.
If the questions says 'simplify' it is a reasonable to assume that it can be done in that instance.
