Question

So sánh :

a) A= \(\sqrt{\dfrac{37}{4}-\sqrt{49+12\sqrt{5}}}\) ; B= \(\sqrt{5}-\dfrac{3}{2}\)

b)C=\(\dfrac{16\sqrt{36}-20\sqrt{48}+10\sqrt{3}}{\sqrt{12}}\) và B =\(16\sqrt{3}-36\)

c) A=\(\sqrt{2015}+\sqrt{2017}\) và B=\(2\sqrt{2016}\)

Answer 1

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\(A=\sqrt{\dfrac{37}{4}-\sqrt{49+12\sqrt{5}}}\)

\(=\sqrt{\dfrac{37}{4}-\sqrt{\left(3\sqrt{5}+2\right)^2}}\)

\(=\sqrt{\dfrac{29}{4}-3\sqrt{5}}\)

\(=\sqrt{\dfrac{29-12\sqrt{5}}{4}}\)

\(=\sqrt{\dfrac{\left(2\sqrt{5}-3\right)^2}{4}}\)

\(=\dfrac{\sqrt{5}}{2}-\dfrac{3}{4}\)

\(=\dfrac{1}{2}\left(\sqrt{5}-\dfrac{3}{2}\right)\)

\(>\sqrt{5}-\dfrac{3}{2}=B\)

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\(C=\dfrac{16\sqrt{36}-20\sqrt{48}+10\sqrt{3}}{\sqrt{12}}\)

\(=\dfrac{96-80\sqrt{3}+10\sqrt{3}}{\sqrt{12}}\)

\(=\dfrac{96-70\sqrt{3}}{2\sqrt{3}}\)

\(=16\sqrt{3}-35\)

\(>16\sqrt{3}-36=B\)

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