\(\sqrt{11+6\sqrt{2}}\)-\(\sqrt{11-6\sqrt{2}}\)rút gọn
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\(P=\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}.\)
\(=\frac{2+\sqrt{3}}{\frac{\sqrt{2}.\sqrt{2}+\sqrt{2}.\sqrt{2+\sqrt{3}}}{\sqrt{2}}}\)\(+\frac{2-\sqrt{3}}{\frac{\sqrt{2}.\sqrt{2}-\sqrt{2}.\sqrt{2-\sqrt{3}}}{\sqrt{2}}}\)
\(=\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{4+\sqrt{3}}}\)\(+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}\)\(+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{3}+1}+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{3}+1}\)
Sai chỗ nào ý nhỉ
\(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\)
= \(\sqrt{7}-1-\left(1+\sqrt{7}\right)\)
\(=\sqrt{7}-1-\sqrt{7}-1\)
= -2
1. \(2\sqrt{5}-5\sqrt{20}+\sqrt{80}\)
= \(2\sqrt{5}-5.2\sqrt{5}+4\sqrt{5}\)
= \(2\sqrt{5}-10\sqrt{5}+4\sqrt{5}\)
= \(-4\sqrt{5}\)
2. B = \(\frac{1}{\sqrt{5}-2}-\sqrt{6-2\sqrt{5}}\)
= \(\frac{1}{\sqrt{5}-2}-\sqrt{1^2-2\sqrt{5}+\sqrt{15}^2}\)
= \(\frac{1}{\sqrt{5}-2}-\sqrt{\left(1-\sqrt{5}\right)^2}\)
= \(\frac{1}{\sqrt{5}-2}-\left|1-\sqrt{5}\right|\)
= \(\frac{1}{\sqrt{5}-2}-\sqrt{5}+1\left(\sqrt{5}>1\right)\)
= \(\frac{1}{\sqrt{5}-2}-\frac{\left(\sqrt{5}+1\right)\left(\sqrt{5}-2\right)}{\sqrt{5}-2}=\frac{1-5+2\sqrt{5}+\sqrt{5}-2}{\sqrt{5}-2}\)
= \(\frac{-6+3\sqrt{5}}{\sqrt{5}-2}=\frac{3\left(\sqrt{5}-2\right)}{\sqrt{5}-2}=3\)
\(x=2\sqrt{2}+3=2+2\sqrt{2}+1=\left(\sqrt{2}+1\right)^2\)
Thay vào A ta được:
\(A=\frac{\left(\sqrt{2}+1\right)^2-1}{\sqrt{\left(\sqrt{2}+1\right)^2}}=\frac{\left(\sqrt{2}+1+1\right)\left(\sqrt{2}+1-1\right)}{\sqrt{2}+1}\)
\(=\frac{\sqrt{2}\left(\sqrt{2}+2\right)}{\sqrt{2}+1}=\frac{2+2\sqrt{2}}{\sqrt{2}+1}=\frac{\left(2+2\sqrt{2}\right)\left(\sqrt{2}-1\right)}{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}\)
\(\frac{2\sqrt{2}-2+4-2\sqrt{2}}{2-1}=2\)
Ta có \(\frac{a^3}{b^2+1}=\frac{a^3}{b^2+ab}=\frac{a^3}{b\left(a+b\right)}\)
Áp dụng BĐT cosi
\(\frac{a^3}{b\left(a+b\right)}+\frac{b}{2}+\frac{a+b}{4}\ge\frac{3}{2}a\)
TT \(\frac{b^3}{a\left(a+b\right)}+\frac{a}{2}+\frac{a+b}{4}\ge\frac{3}{2}b\)
=> \(VT\ge\frac{1}{2}\left(a+b\right)\ge\sqrt{ab}=1\)
Dấu bằng xảy ra khi a=b=1
Sửa đề \(A=\frac{x^2}{x+y^2}+\frac{y^2}{y+z^2}+\frac{z^2}{z+x^2}\)
Ta có \(\frac{x^2}{x+y^2}=\frac{x^2+xy^2-xy^2}{x+y^2}=x-\frac{xy^2}{x+y^2}\)
Mà \(x+y^2\ge2y\sqrt{x}\)
=> \(\frac{x^2}{x+y^2}\ge x-\frac{\sqrt{x}}{2}\ge\frac{3x}{4}-\frac{1}{4}\)
=> \(VT\ge\frac{3}{4}\left(x+y+z\right)-\frac{3}{4}=\frac{9}{4}-\frac{3}{4}=\frac{3}{2}\)
Min A=3/2 khi x=y=z=1
Ta có: \(\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}=\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}\)
\(=\left|3+\sqrt{2}\right|-\left|3-\sqrt{2}\right|\)
\(=3+\sqrt{2}-3+\sqrt{2}\)
\(=2\sqrt{2}\)
\(\sqrt{11+6\sqrt{2}}-\sqrt{11-6\sqrt{2}}\)
\(=\sqrt{9+2.3\sqrt{2}+2}-\sqrt{9-2.3\sqrt{2}+2}\)
\(=3+\sqrt{2}-3+\sqrt{2}\)
\(=2\sqrt{2}\)