$$\sqrt{4+2\sqrt3}=1+\sqrt{3}$$
What steps would one take to get $1+\sqrt{3}$? Squaring the right side obviously gets to the expression under the square root on the left side, but I don't know how to go the other way.
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1Cf. this answer – J. W. Tanner Oct 28 '19 at 17:50
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$(a+b)+2\sqrt{a\times b}=(\sqrt a+\sqrt b)^2$ – mathlove Oct 28 '19 at 17:55
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See this answer in the linked dupe for a very simple algorithm to denest such radicals. It has links to many worked examples. – Bill Dubuque Oct 28 '19 at 18:11
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Hint:
Try to write $$\sqrt{4+2\sqrt3}=a+b\sqrt 3$$ and now square it. Try with assumption that $a,b$ are integers. You get $$4+2\sqrt3 = a^2+3b^2 +2ab\sqrt{3}$$
So try with $a^2+3b^2 = 4$ and $2ab = 2$.
nonuser
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Make the ansatz $$\sqrt{4+2\sqrt{3}}=a+b\sqrt{3}$$. For your work: It is $$(1+\sqrt{3})^2=1+3+2\sqrt{3}$$
Dr. Sonnhard Graubner
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