We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
160
5
avatar+221 

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

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

 Mar 29, 2019
 #1
avatar+104444 
+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+4323 
+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+104444 
+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+52 
+1

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

neworleans06  Mar 30, 2019
 #5
avatar+104444 
+1

Yep, that is what Tertre means :)

Melody  Mar 30, 2019

8 Online Users