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Simplify \(\large \sqrt{36 + 16\sqrt5 }\)

 Jun 29, 2020
 #1
avatar+732 
+1

\(\sqrt{36+16\sqrt{5}}\)

\(=\sqrt{4(9+4\sqrt{5})}\)         (factor the 36 and 16 in the square root)

\(=2\sqrt{9+4\sqrt{5}}\)             (take out the 4 in the square root)

\(=2\sqrt{9+2\sqrt{20}}\)           (rewrote)

 

now let's just focus of the \(9+2\sqrt{20}\) for now

\(9+2\sqrt{20}=4+5+2\sqrt{20}\)

\(4+5+2\sqrt{20}=(\sqrt{4})^2+(\sqrt{5})^2+2\cdot\sqrt{4}\cdot\sqrt{5}\) 

***you get this by using the formula: \(a^2+b^2+2ab=(a+b)^2\)***

 

so, \(9+2\sqrt{20}=(\sqrt{4}+{\sqrt{5}})^2\)

 

now we can plug that back into the original equation.

\(=2\sqrt{9+2\sqrt{20}}=2\sqrt{(\sqrt{4}+\sqrt{5})^2}\)

\(=2(\sqrt{4}+\sqrt{5})\)

\(=2(2+\sqrt{5})\)

\(=\boxed{4+2\sqrt{5}}\)

 

 

sorry if this was unclear, i was in a bit of a hurry 

 

 

\(\)

 Jun 29, 2020
edited by lokiisnotdead  Jun 29, 2020
edited by lokiisnotdead  Jun 29, 2020
 #2
avatar+549 
+3

this was very clear! but you could add that there is a formula: \(a^2+2ab+b^2=(a+b)^2\)

amazingxin777  Jun 29, 2020
 #3
avatar+732 
+1

thanks for the suggestion! i edited it and added that in! :)

lokiisnotdead  Jun 29, 2020

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