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1) Fully simplify \(\sqrt{49-20\sqrt{6}}\)

2) Fully simplify \(\sqrt {14 + 8\sqrt {3}}\)

 Mar 29, 2019
 #1
avatar+109795 
+2

\(\sqrt{49-20\sqrt6}\\ =\sqrt{25+24-20\sqrt6}\\ \text{This might be helpful because 25 is a square and 24 is a multiple of 6}\\ =\sqrt{25-20\sqrt6+4*6}\\ =\sqrt{5^2-20\sqrt6+(2\sqrt6)^2}\\ =\sqrt{5^2-2*5*2\sqrt6+(2\sqrt6)^2}\\ =\sqrt{(5-2\sqrt6)^2}\\ =|5-2\sqrt6|\\ =5-2\sqrt6\\\)

 

Now maybe you can attempt the second one.   

Start by factoring out the 2.

 Mar 30, 2019
 #2
avatar+4586 
+3

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 \(a-b=\boxed{5-2\sqrt{6}}.\)

.
 Mar 30, 2019
 #3
avatar+109795 
+1

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.

Melody  Mar 30, 2019
 #4
avatar+54 
+1

Nice solution tertre, but I think you meant to say \(a^2+b^2=49?\)

neworleans06  Mar 30, 2019
 #5
avatar+109795 
+1

Yep, that is what Tertre means :)

Melody  Mar 30, 2019

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