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# square roots

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302
5

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

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

Mar 29, 2019

#1
+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
+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 $$a-b=\boxed{5-2\sqrt{6}}.$$

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Mar 30, 2019
#3
+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
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Nice solution tertre, but I think you meant to say $$a^2+b^2=49?$$

neworleans06  Mar 30, 2019
#5
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Yep, that is what Tertre means :)

Melody  Mar 30, 2019