1) Khi x = 49 thì:

\(A=\frac{4\sqrt{49}}{\sqrt{49}-1}=\frac{4\cdot7}{7-1}=\frac{28}{6}=\frac{14}{3}\)

2) Ta có:

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

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

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

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

c) \(P=A\div B=\frac{4\sqrt{x}}{\sqrt{x}-1}\div\frac{\sqrt{x}+1}{\sqrt{x}-1}=\frac{4\sqrt{x}}{\sqrt{x}+1}\)

Ta có: \(P\left(\sqrt{x}+1\right)=x+4+\sqrt{x-4}\)

\(\Leftrightarrow\frac{4\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+1}=x+4+\sqrt{x-4}\)

\(\Leftrightarrow4\sqrt{x}=x+4+\sqrt{x-4}\)

\(\Leftrightarrow\left(\sqrt{x}-2\right)^2+\sqrt{x-4}=0\)

Mà \(VT\ge0\left(\forall x\ge0,x\ne1\right)\)

\(\Rightarrow\hept{\begin{cases}\left(\sqrt{x}-2\right)^2=0\\\sqrt{x-4}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}\sqrt{x}=2\\x-4=0\end{cases}}\Rightarrow x=4\)

Vậy x = 4

Em gửi ảnh ạ !

Em gửi ảnh trên ạ !!!!!

                      Bài làm :

1) Khi x=9 ; giá trị của A là :

\(A=\frac{\sqrt{9}}{\sqrt{9}+2}=\frac{3}{3+2}=\frac{3}{5}\)

2) Ta có :

\(B=...\)

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

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

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

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

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

3) Ta có :

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

Xét :

\(\frac{A}{B}+1=\frac{4}{\sqrt{x+2}}>0\Rightarrow\frac{A}{B}>-1\)

=> Điều phải chứng minh

1, thay x=9(TMĐKXĐ) vào A ta đk:

A=\(\dfrac{\sqrt{9}}{\sqrt{9}-2}=3\)

vậy khi x=9 thì A =3

2,với x>0,x≠4 ta đk:

B=\(\dfrac{x}{x-4}+\dfrac{1}{\sqrt{x}-2}+\dfrac{1}{\sqrt{x}+2}=\dfrac{x+\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\dfrac{x+2\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}}{\sqrt{x}-2}\)

vậy B=\(\dfrac{\sqrt{x}}{\sqrt{x}-2}\)

3,\(\dfrac{A}{B}>-1\) (x>0,x≠4)

⇒\(\dfrac{\sqrt{x}}{\sqrt{x}+2}:\dfrac{\sqrt{x}}{\sqrt{x}-2}>-1\Leftrightarrow\dfrac{\sqrt{x}}{\sqrt{x}+2}.\dfrac{\sqrt{x}-2}{\sqrt{x}}>-1\Leftrightarrow\dfrac{\sqrt{x}-2}{\sqrt{x}+2}>-1\)

⇒\(\sqrt{x}-2>-1\) (vì \(\sqrt{x}+2>0\))

⇔\(\sqrt{x}>1\)⇔x=1 (TM)

vậy x=1 thì \(\dfrac{A}{B}>-1\) với x>0 và x≠4

a: \(P=\dfrac{x-\sqrt{x}-1-\sqrt{x}+1}{x-1}\cdot\dfrac{4\left(\sqrt{x}-2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)^2}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)\cdot4\left(\sqrt{x}-2\right)}{\sqrt{x}\left(x-1\right)}=\dfrac{4}{x-1}\)

Để P nguyên dương thì x-1 thuộc {1;4;2}

=>x thuộc {2;5;3}

b: x+y+z=0

=>x=-y-z; y=-x-z; z=-x-y

\(P=\dfrac{x^2}{y^2+z^2-\left(y+z\right)^2}+\dfrac{y^2}{z^2+x^2-\left(x+z\right)^2}+\dfrac{z^2}{x^2+y^2-\left(x+y\right)^2}\)

\(=\dfrac{x^2}{-2yz}+\dfrac{y^2}{-2xz}+\dfrac{z^2}{-2xy}\)

\(=\dfrac{x^3+y^3+z^3}{2xyz}\cdot\left(-1\right)\)

\(=-\dfrac{\left(x+y\right)^3+z^3-3xy\left(x+y\right)}{2xyz}\)

\(=-\dfrac{\left(-z\right)^3+z^3-3xy\cdot\left(-z\right)}{2xyz}=-\dfrac{3}{2}\)

1:

ĐKXĐ: x≠4

Ta có: \(x=\sqrt{7-4\sqrt{3}}+\sqrt{7+4\sqrt{3}}\)

\(=\sqrt{3-2\cdot\sqrt{3}\cdot2+4}+\sqrt{3+2\cdot\sqrt{3}\cdot2+4}\)

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

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

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

\(=4\)(ktm ĐKXĐ)

Vậy: Khi x=4 thì A không có giá trị

2: Ta có: P=A+B

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

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

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

\(\Leftrightarrow P=\frac{3x-6\sqrt{x}}{\left(x+1\right)\left(\sqrt{x}-2\right)}=\frac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(x+1\right)\left(\sqrt{x}-2\right)}\)

\(\Leftrightarrow P=\frac{3\sqrt{x}}{x+1}\)

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

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

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

Để C nguyên \(\Rightarrow\sqrt{x}+2=Ư\left(2\right)=2\)

\(\Rightarrow\sqrt{x}=0\Rightarrow x=0\)

a/ ĐKXĐ:...

\(B=\left(\frac{\sqrt{x}+\sqrt{x}+2}{x-4}\right).\frac{x-4}{\sqrt{x}+2}=\frac{2\sqrt{x}+2}{\sqrt{x}+2}\)

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

\(C=\frac{\sqrt{x}+2}{\sqrt{x}-2}.\frac{2\sqrt{x}+2-2\sqrt{x}-4}{\sqrt{x}+2}=\frac{-2}{\sqrt{x}-2}\)

Để C đạt GT nguyên

\(\Leftrightarrow\sqrt{x}-2\inƯ_{\left(-2\right)}=\left\{\pm2;\pm1\right\}\)

\(\left[{}\begin{matrix}x=0\\x=9\\x=1\\x=16\end{matrix}\right.\)

cho hỏi là mẫu biểu thức A là\(\sqrt{x}-3\) hay\(\sqrt{x-3}\)

là\(\sqrt{x}-3\)mình ghi nhầm