Question

a,\(\dfrac{1}{2+\sqrt{3}}+\dfrac{\sqrt{2}}{\sqrt{6}}-\dfrac{2}{3+\sqrt{3}}\)

b,\(\sqrt{x-3+2\sqrt{x-4}}\)

c,\(\sqrt{2x+4\sqrt{2x-4}}\)

d,\(\sqrt{x-\sqrt{2x-1}}\)

e,\(\sqrt{x+6\sqrt{x-9}}-\sqrt{x-9}\)

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Answer 1

Lời giải:
a)

\(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}+\frac{\sqrt{3}}{3}-\frac{2(3-\sqrt{3})}{(3+\sqrt{3})(3-\sqrt{3})}\)

\(=\frac{2-\sqrt{3}}{4-3}+\frac{\sqrt{3}}{3}-\frac{2(3-\sqrt{3})}{3^2-3}\)

\(=2-\sqrt{3}+\frac{\sqrt{3}}{3}-\frac{3-\sqrt{3}}{3}=\frac{6-3\sqrt{3}}{3}+\frac{2\sqrt{3}-3}{3}=\frac{3-\sqrt{3}}{3}\)

b)

\(\sqrt{x-3+2\sqrt{x-4}}=\sqrt{(x-4)+2\sqrt{x-4}+1}=\sqrt{(\sqrt{x-4}+1)^2}=|\sqrt{x-4}+1|=\sqrt{x-4}+1\)

c)

\(\sqrt{2x+4\sqrt{2x-4}}=\sqrt{(2x-4)+2.2\sqrt{2x-4}+2^2}\)

\(=\sqrt{(\sqrt{2x-4}+2)^2}=|\sqrt{2x-4}+2|=\sqrt{2x-4}+2\)

d)

\(\sqrt{x-\sqrt{2x-1}}=\frac{1}{\sqrt{2}}\sqrt{2x-2\sqrt{2x-1}}=\frac{1}{\sqrt{2}}\sqrt{(2x-1)-2\sqrt{2x-1}+1}\)

\(=\frac{1}{\sqrt{2}}\sqrt{(\sqrt{2x-1}-1)^2}=\frac{|\sqrt{2x-1}-1|}{\sqrt{2}}\)

e)

\(\sqrt{x+6\sqrt{x-9}}-\sqrt{x-9}=\sqrt{(x-9)+2.3\sqrt{x-9}+3^2}-\sqrt{x-9}\)

\(=\sqrt{(\sqrt{x-9}+3)^2}-\sqrt{x-9}=|\sqrt{x-9}+3|-\sqrt{x-9}\)

\(=\sqrt{x-9}+3-\sqrt{x-9}=3\)

Answer 2

Lời giải:
a)

\(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}=\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}+\frac{\sqrt{3}}{3}-\frac{2(3-\sqrt{3})}{(3+\sqrt{3})(3-\sqrt{3})}\)

\(=\frac{2-\sqrt{3}}{4-3}+\frac{\sqrt{3}}{3}-\frac{2(3-\sqrt{3})}{3^2-3}\)

\(=2-\sqrt{3}+\frac{\sqrt{3}}{3}-\frac{3-\sqrt{3}}{3}=\frac{6-3\sqrt{3}}{3}+\frac{2\sqrt{3}-3}{3}=\frac{3-\sqrt{3}}{3}\)

b)

\(\sqrt{x-3+2\sqrt{x-4}}=\sqrt{(x-4)+2\sqrt{x-4}+1}=\sqrt{(\sqrt{x-4}+1)^2}=|\sqrt{x-4}+1|=\sqrt{x-4}+1\)

c)

\(\sqrt{2x+4\sqrt{2x-4}}=\sqrt{(2x-4)+2.2\sqrt{2x-4}+2^2}\)

\(=\sqrt{(\sqrt{2x-4}+2)^2}=|\sqrt{2x-4}+2|=\sqrt{2x-4}+2\)

d)

\(\sqrt{x-\sqrt{2x-1}}=\frac{1}{\sqrt{2}}\sqrt{2x-2\sqrt{2x-1}}=\frac{1}{\sqrt{2}}\sqrt{(2x-1)-2\sqrt{2x-1}+1}\)

\(=\frac{1}{\sqrt{2}}\sqrt{(\sqrt{2x-1}-1)^2}=\frac{|\sqrt{2x-1}-1|}{\sqrt{2}}\)

e)

\(\sqrt{x+6\sqrt{x-9}}-\sqrt{x-9}=\sqrt{(x-9)+2.3\sqrt{x-9}+3^2}-\sqrt{x-9}\)

\(=\sqrt{(\sqrt{x-9}+3)^2}-\sqrt{x-9}=|\sqrt{x-9}+3|-\sqrt{x-9}\)

\(=\sqrt{x-9}+3-\sqrt{x-9}=3\)