Ta có: \(\frac{1}{3}\left(\sqrt{6}+\sqrt{5}\right)^2-\frac{1}{4}\sqrt{120}-2\sqrt{\frac{15}{2}}\)

\(=\frac{1}{3}\left(11+2\sqrt{30}\right)-\frac{\sqrt{30}}{2}-\sqrt{30}\)

\(=\frac{11}{3}+\frac{2}{3}\sqrt{30}-\frac{\sqrt{30}}{2}-\sqrt{30}\)

\(=\frac{11}{3}-\frac{5}{6}\sqrt{30}\)

\(=\frac{22-5\sqrt{30}}{6}\)

Ta có: \(\left(\frac{1}{2}\sqrt{\frac{2}{3}}-\frac{3}{4}\sqrt{54}+\frac{1}{3}\sqrt{\frac{8}{3}}\right)\div\sqrt{\frac{81}{6}}\)

\(=\left(\frac{\sqrt{6}}{6}-\frac{9\sqrt{6}}{4}+\frac{2\sqrt{6}}{9}\right)\div\frac{3\sqrt{6}}{2}\)

\(=-\frac{67\sqrt{6}}{36}\cdot\frac{2}{3\sqrt{6}}\)

\(=-\frac{67}{54}\)

1) \(\left(\sqrt{6}-\sqrt{8}\right)\left(\sqrt{6}+\sqrt{8}\right)\)

\(=\left(\sqrt{6}\right)^2-\left(\sqrt{8}\right)^2\)

\(=6-8=-2\)

2) \(\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)\)

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

\(=9-5=4\)

3) \(\sqrt{7-4\sqrt{3}}+\sqrt{7+4\sqrt{3}}\)

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

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

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

4) Xét ta thấy: \(2\sqrt{3}=\sqrt{12}< \sqrt{16}=4\)

=> \(2\sqrt{3}-4< 0\) => vô lý không tm đk căn

5) Đặt \(A=\sqrt{4\sqrt{6}+11}-\sqrt{11-4\sqrt{6}}\)

\(\Rightarrow A^2=\left(\sqrt{11+4\sqrt{6}}-\sqrt{11-4\sqrt{6}}\right)^2\)

\(=11+4\sqrt{6}-2\sqrt{\left(11+4\sqrt{6}\right)\left(11-4\sqrt{6}\right)}+11-4\sqrt{6}\)

\(=22-2\sqrt{121-96}\)

\(=22-2\sqrt{5}\)

=> \(A=\sqrt{22-2\sqrt{5}}\)

6) Đặt \(B=\sqrt{10+2\sqrt{11}}-\sqrt{10-2\sqrt{11}}\)

\(\Leftrightarrow B^2=\left(\sqrt{10+2\sqrt{11}}-\sqrt{10-2\sqrt{11}}\right)^2\)

\(=10+2\sqrt{11}-2\sqrt{\left(10+2\sqrt{11}\right)\left(10-2\sqrt{11}\right)}+10-2\sqrt{11}\)

\(=20-2\sqrt{100-44}\)

\(=20-4\sqrt{14}\)

=> \(B=\sqrt{20-4\sqrt{14}}\)

\(ĐKXĐ:\hept{\begin{cases}a>0\\a\ne1\end{cases}}\)

\(P=\frac{2a+4}{a\sqrt{a}-1}+\frac{\sqrt{a}+2}{a+\sqrt{a}+1}-\frac{2}{\sqrt{a}-1}\)

\(=\frac{2a+4+\left(\sqrt{a}+2\right)\left(\sqrt{a}-1\right)-2\left(a+\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\)

\(=\frac{2a+4+a+\sqrt{a}-2-2a-2\sqrt{a}-2}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\)

\(=\frac{a-\sqrt{a}}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\)

\(=\frac{\sqrt{a}}{a+\sqrt{a}+1}\)

Ta có:

\(P=\frac{2a+4}{a\sqrt{a}-1}+\frac{\sqrt{a}+2}{a+\sqrt{a}+1}-\frac{2}{\sqrt{a}-1}\)

\(P=\frac{2a+4+\left(\sqrt{a}+2\right)\left(\sqrt{a}-1\right)-2\left(a+\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\)

\(P=\frac{2a+4+a+\sqrt{a}-2-2a-2\sqrt{a}-2}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\)

\(P=\frac{a-\sqrt{a}}{\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\)

\(P=\frac{\sqrt{a}}{a+\sqrt{a}+1}\)

trả lời nhanh hộ t nhé cc :)

\(\frac{5\left(\sqrt{6}-1\right)\left(\sqrt{6}-1\right)}{\left(\sqrt{6}+1\right)\left(\sqrt{6}-1\right)}+\frac{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}+\sqrt{\left(\sqrt{2}\right)^2-2\sqrt{2}+1}\)

\(=\frac{5\left(\sqrt{6}-1\right)^2}{5}-\frac{\left(\sqrt{2}-\sqrt{3}\right)^2}{1}+\sqrt{\left(\sqrt{2}-1\right)^2}\)

\(=\left(\sqrt{6}-1\right)^2-\left(\sqrt{2}-\sqrt{3}\right)^2+\left(\sqrt{2}-1\right)\)

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

Ta có: \(C=\dfrac{\sqrt{2+\sqrt{3}}}{2}:\left(\dfrac{\sqrt{2+\sqrt{3}}}{2}-\dfrac{2}{\sqrt{6}}+\dfrac{\sqrt{2+\sqrt{3}}}{2\sqrt{3}}\right)\)

\(=\dfrac{\sqrt{4+2\sqrt{3}}}{2\sqrt{2}}:\left(\dfrac{\sqrt{6+3\sqrt{3}}}{2\sqrt{3}}-\dfrac{2\sqrt{2}}{2\sqrt{3}}+\dfrac{\sqrt{2+\sqrt{3}}}{2\sqrt{3}}\right)\)

\(=\dfrac{\sqrt{3}+1}{2\sqrt{2}}:\left(\dfrac{\sqrt{12+6\sqrt{3}}-4+\sqrt{4+2\sqrt{3}}}{2\sqrt{6}}\right)\)

\(=\dfrac{\sqrt{3}+1}{2\sqrt{2}}:\dfrac{3+\sqrt{3}-4+\sqrt{3}+1}{2\sqrt{6}}\)

\(=\dfrac{\sqrt{3}+1}{2\sqrt{2}}\cdot\dfrac{2\sqrt{6}}{2\sqrt{3}}\)

\(=\dfrac{\sqrt{3}+1}{2}\)

ok

\(\sqrt{-3x^3+5x+14}+\sqrt{-5x^3+6x+28}=\left(4-2x-x^2\right)\sqrt{2-x}\) (ĐKXĐ: \(x\in R,x\le2\))

\(\Leftrightarrow\sqrt{\left(2-x\right)\left(3x^2+6x+7\right)}+\sqrt{\left(2-x\right)\left(5x^2+10x+14\right)}-\left(4-2x-x^2\right)\sqrt{2-x}=0\)

\(\Leftrightarrow\sqrt{2-x}\left(\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}-4+2x+x^2\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=2\left(tm\right)\\\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}=4-2x-x^2\left(1\right)\end{cases}}\)

Pt \(\left(1\right)\Leftrightarrow\sqrt{3\left(x+1\right)^2+4}+\sqrt{5\left(x+1\right)^2+9}=-\left(x+1\right)^2+5\left(2\right)\)

Ta có: \(\left(x+1\right)^2\ge0\Rightarrow\sqrt{2\left(x+1\right)^2+4}\ge\sqrt{4}=2\)

Tương tự: \(\sqrt{5\left(x+1\right)^2+9}\ge3\). Từ đó: \(VT_{\left(2\right)}\)\(\ge2+3=5\)

Mà \(VP_{\left(2\right)}=-\left(x+1\right)^2+5\le5\) nên dấu "=" xảy ra \(\Leftrightarrow\left(x+1\right)^2=0\Leftrightarrow x=-1\)(tm)

Vậy tập nghiệm của pt cho là \(S=\left\{2;-1\right\}.\)

Câu 1: 

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