Đk: x \(\ge\)0; x \(\ne\)1; x \(\ne\)4; x \(\ne\)16

A = \(\frac{x\sqrt{x}-4x-\sqrt{x}+4}{2x\sqrt{x}-14x+28\sqrt{x}-16}=\frac{\sqrt{x}\left(x-1\right)-4\left(x-1\right)}{2\left(x\sqrt{x}-8\right)-14\sqrt{x}\left(\sqrt{x}-2\right)}\)

A = \(\frac{\left(\sqrt{x}-4\right)\left(x-1\right)}{2\left[\left(\sqrt{x}-2\right)\left(x+2\sqrt{x}+4\right)-7\sqrt{x}\left(\sqrt{x}-2\right)\right]}\)

A = \(\frac{\left(\sqrt{x}-4\right)\left(x-1\right)}{2\left(\sqrt{x}-2\right)\left(x-5\sqrt{x}+4\right)}=\frac{\left(\sqrt{x}-4\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}-4\right)}=\frac{\sqrt{x}+1}{2\left(\sqrt{x}-2\right)}\)

Số thực thôi không phải dương à:vv?

Dương nhé bạn.

\(\sqrt{x}+\sqrt{1-x}=1\)   

\(\left(ĐK:0\le x\le1\right)\)

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

\(2\sqrt{x\left(1-x\right)}=1-1\)   

\(2\sqrt{x\left(1-x\right)}=0\)   

\(\sqrt{x\left(1-x\right)}=0:2\)   

\(\sqrt{x\left(1-x\right)}=0\)   

\(x\left(1-x\right)=0^2\)   

\(x\left(1-x\right)=0\)   

\(\orbr{\begin{cases}x=0\\1-x=0\end{cases}}\)   

\(\orbr{\begin{cases}x=0\left(n\right)\\x=1\left(n\right)\end{cases}}\)

\(8=x^2+y^2-xy\ge x^2+y^2-\frac{x^2+y^2}{2}=\frac{x^2+y^2}{2}\)

\(\Rightarrow x^2+y^2\le8.2=16\)

Dấu \(=\)khi \(x=y=2\sqrt{2}\).

\(x^2+y^2=8+xy\ge8\)

Dấu \(=\)khi \(\hept{\begin{cases}x=0\\y=2\sqrt{2}\end{cases}}\)hoặc \(\hept{\begin{cases}x=2\sqrt{2}\\y=0\end{cases}}\).

giá trị nhỏ nhất của M là 0
khi x=0 và y=0

\(\left(3\sqrt{2}+\sqrt{6}\right)\sqrt{6-3\sqrt{3}}=\left(3+\sqrt{3}\right)\sqrt{12-6\sqrt{3}}=\left(3+\sqrt{3}\right)\sqrt{9-2.3.\sqrt{3}+3}\)

\(=\left(3+\sqrt{3}\right)\sqrt{3^2-2.3.\sqrt{3}+\left(\sqrt{3}\right)^2}=\left(3+\sqrt{3}\right)\sqrt{\left(3-\sqrt{3}\right)^2}\)

\(=\left(3+\sqrt{3}\right)\left|3-\sqrt{3}\right|=\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)=6\)

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

\(=\frac{a+2\sqrt{ab}+b-a+2\sqrt{ab}-b}{a-b}=\frac{4\sqrt{ab}}{a-b}\)

2) \(x-4-\sqrt{16-8x^2+x^4}=x-4-\sqrt{\left(x^2-4\right)^2}=x-4-\left|x^2-4\right|\)

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

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

4) \(\frac{a+b-2\sqrt{ab}}{\sqrt{a}-\sqrt{b}}:\frac{1}{\sqrt{a}+\sqrt{b}}=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\sqrt{a}-\sqrt{b}}\cdot\left(\sqrt{a}+\sqrt{b}\right)\)

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

5) \(\frac{a-3\sqrt{a}}{\sqrt{a}-3}-\frac{a+4\sqrt{a}+3}{\sqrt{a}+3}=\frac{\sqrt{a}\left(\sqrt{a}-3\right)}{\sqrt{a}-3}-\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}+3\right)}{\sqrt{a}+3}\)

\(=\sqrt{a}-\sqrt{a}-1=-1\)

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

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

7) \(\frac{x\sqrt{y}+y\sqrt{x}}{\sqrt{xy}}:\frac{\sqrt{x}-\sqrt{y}}{x-y}=\frac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}}:\frac{\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\)

\(=\left(\sqrt{x}+\sqrt{y}\right)^2\)

8) \(\frac{\sqrt{a}+3}{\sqrt{a}-2}-\frac{\sqrt{a}-1}{\sqrt{a}+2}+\frac{4\sqrt{a}-4}{4-a}=\frac{\left(\sqrt{a}+3\right)\left(\sqrt{a}+2\right)-\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)-4\sqrt{a}+4}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)

=\(\frac{a+5\sqrt{a}+6-a+3\sqrt{a}-2-4\sqrt{a}+4}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}=\frac{4\sqrt{a}+8}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}=\frac{4}{\sqrt{a}-2}\)

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

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

10) \(\left(\frac{2+\sqrt{x}}{2-\sqrt{x}}+\frac{2-\sqrt{x}}{2+\sqrt{x}}-\frac{4x}{x-4}\right):\frac{x-6\sqrt{x}+9}{\left(2-\sqrt{x}\right)\left(\sqrt{x}-3\right)}\)

\(=\left(\frac{\left(2+\sqrt{x}\right)^2-\left(2-\sqrt{x}\right)^2+4x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}\right):\frac{\left(\sqrt{x}-3\right)^2}{\left(2-\sqrt{x}\right)\left(\sqrt{x}-3\right)}\)

\(=\frac{x+4\sqrt{x}+4-x+4\sqrt{x}-4+4x}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\cdot\frac{2-\sqrt{x}}{\sqrt{x}-3}\)

\(=\frac{4x+8\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+2\right)}=\frac{4\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+2\right)}=\frac{4\sqrt{x}}{\sqrt{x}-3}\)

11) \(\left(\frac{2\sqrt{x}+x}{x\sqrt{x}-1}-\frac{1}{\sqrt{x}-1}\right):\left(\frac{\sqrt{x}+2}{x+\sqrt{x}+1}\right)=\frac{2\sqrt{x}+x-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\frac{x+\sqrt{x}+1}{\sqrt{x}+2}\)

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

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

\(=\left(\sqrt{x}+1\right)^2\)

a) Ta có:

\(\sqrt{\frac{289}{225}}=\sqrt{\frac{\sqrt{289}}{\sqrt{225}}}=\sqrt{\frac{17^2}{15^2}}=\frac{17}{15}\)

b) Ta có:

\(\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{25}}=\sqrt{\frac{\sqrt{64}}{\sqrt{25}}}=\sqrt{\frac{8^2}{5^2}}=\frac{8}{5}\)

c) Ta có:

\(\sqrt{\frac{0,25}{9}}=\sqrt{\frac{\sqrt{0,25}}{\sqrt{9}}}=\sqrt{\frac{0,5^2}{3^2}}=\frac{0,5}{3}=\frac{1}{6}\)

d) Ta có:

\(\sqrt{\frac{8,1}{1,6}}=\sqrt{\frac{81.0,1}{16.0,1}}=\sqrt{\frac{81}{16}}=\sqrt{\frac{\sqrt{81}}{\sqrt{16}}}=\sqrt{\frac{9^2}{4^2}}=\frac{9}{4}\)

a)Ta có: \(\sqrt{\frac{289}{225}}=\frac{\sqrt{289}}{\sqrt{225}}=\frac{17}{15}\)

b) Ta có: \(\sqrt{2\frac{14}{25}}=\sqrt{\frac{64}{25}}=\frac{\sqrt{64}}{\sqrt{25}}=\frac{8}{5}\)

c) Ta có: \(\sqrt{\frac{0,25}{9}}=\frac{\sqrt{0,25}}{\sqrt{9}}=\frac{0,5}{3}=\frac{1}{6}\)

d)Ta có : \(\sqrt{\frac{8,1}{1,6}}=\frac{\sqrt{8,1}}{\sqrt{1,6}}=\frac{\sqrt{8,1}.100}{\sqrt{1,6}.100}=\frac{\sqrt{81}}{\sqrt{16}}=\frac{9}{4}\)