ai lm ơn giúp mk với

HELP ME

\(z=\frac{\sqrt[3]{9\sqrt{3}+11\sqrt{2}}+\sqrt[3]{9\sqrt{3}-11\sqrt{2}}}{2}\) 

= \(\frac{\sqrt[3]{3\sqrt{3}+9\sqrt{2}+6\sqrt{3}+2\sqrt{2}}+\sqrt[3]{3\sqrt{3}-9\sqrt{2}+6\sqrt{3}-2\sqrt{2}}}{2}\) 

= \(\frac{\sqrt[3]{\left(\sqrt{3}+\sqrt{2}\right)^3}+\sqrt[3]{\left(\sqrt{3}-\sqrt{2}\right)^3}}{2}\) 

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

Lời giải:
a.

\(=\sqrt{5+2.2\sqrt{5}+2^2}-\sqrt{5-2.2\sqrt{5}+2^2}\)

$=\sqrt{(\sqrt{5}+2)^2}-\sqrt{(\sqrt{5}-2)^2}$

$=|\sqrt{5}+2|-|\sqrt{5}-2|=(\sqrt{5}+2)-(\sqrt{5}-2)=4$

b.

$=\sqrt{3-2.3\sqrt{3}+3^2}+\sqrt{3+2.3.\sqrt{3}+3^2}$

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

$=|\sqrt{3}-3|+|\sqrt{3}+3|$

$=(3-\sqrt{3})+(\sqrt{3}+3)=6$

c.

$=\sqrt{2+2.3\sqrt{2}+3^2}-\sqrt{2-2.3\sqrt{2}+3^2}$

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

$=(\sqrt{2}+3)-(3-\sqrt{2})=2\sqrt{2}$

a) Ta có: \(\sqrt{3-2\sqrt{2}}-\sqrt{11+6\sqrt{2}}\)

\(=\sqrt{2}-1-3-\sqrt{2}\)

=-4

b) Ta có: \(\sqrt{4-2\sqrt{3}}-\sqrt{7-4\sqrt{3}}+\sqrt{19+8\sqrt{3}}\)

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

\(=3\sqrt{3}+1\)

c) Ta có: \(\sqrt{6-2\sqrt{5}}+\sqrt{9+4\sqrt{5}}-\sqrt{14-6\sqrt{5}}\)

\(=\sqrt{5}-1+\sqrt{5}-2-3+\sqrt{5}\)

\(=3\sqrt{5}-6\)

d) Ta có: \(\sqrt{11-4\sqrt{7}}+\sqrt{23-8\sqrt{7}}+\sqrt{\left(-2\right)^6}\)

\(=\sqrt{7}-2+4-\sqrt{7}+8\)

=10

Ta có: \(A=\dfrac{15\sqrt{x}-11}{x+2\sqrt{x}-3}-\dfrac{3\sqrt{x}-2}{\sqrt{x}-1}-\dfrac{2\sqrt{x}+3}{\sqrt{x}+3}\)

\(=\dfrac{15\sqrt{x}-11-3x-9\sqrt{x}+2\sqrt{x}+6-\left(2x-2\sqrt{x}+3\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{-3x+8\sqrt{x}-5-2x-\sqrt{x}+3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{-5x+7\sqrt{x}-2}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}\)

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

\(\Leftrightarrow A-\dfrac{2}{3}=\dfrac{-5\sqrt{x}+2}{\sqrt{x}+3}-\dfrac{2}{3}\)

\(\Leftrightarrow A-\dfrac{2}{3}=\dfrac{-15\sqrt{x}+6-2\sqrt{x}-6}{3\left(\sqrt{x}+3\right)}\)

\(\Leftrightarrow A-\dfrac{2}{3}=\dfrac{-17\sqrt{x}}{3\left(\sqrt{x}+3\right)}\le0\)

\(\Leftrightarrow A\le\dfrac{2}{3}\)

Đặt \(\sqrt[3]{2}=x\Rightarrow2=x^3\Rightarrow x^3+1=3;x^3-1=1\)

\(\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{x-1}=\sqrt[3]{\dfrac{\left(x-1\right)\left(x^2+x+1\right)}{x^2+x+1}}=\sqrt[3]{\dfrac{x^3-1}{x^2+x+1}}\)

\(=\sqrt[3]{\dfrac{1}{x^2+x+1}}=\sqrt[3]{\dfrac{1}{x^2+x+\dfrac{1}{3}\left(x^3+1\right)}}\)

\(=\sqrt[3]{\dfrac{3}{x^3+3x^2+3x+1}}=\sqrt[3]{\dfrac{27}{9\left(x+1\right)^3}}=\dfrac{1}{\sqrt[3]{9}}.\dfrac{3}{x+1}\)

\(=\dfrac{1}{\sqrt[3]{9}}\left(\dfrac{x^3+1}{x+1}\right)=\dfrac{1}{\sqrt[3]{9}}\left(1-x+x^2\right)=\dfrac{1}{\sqrt[3]{9}}\left(1-\sqrt[3]{2}+\sqrt[3]{4}\right)\)

\(=\sqrt[3]{\dfrac{1}{9}}-\sqrt[3]{\dfrac{2}{9}}+\sqrt[3]{\dfrac{4}{9}}\) (đpcm)