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# help

+1
101
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Simplify $$\large \sqrt{36 + 16\sqrt5 }$$

Jun 29, 2020

#1
+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
+600
+4

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
+732
+1

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