Câu 3 : Ta có :

\(\left\{{}\begin{matrix}a^2+1=a^2+ab+bc+ca=\left(a+b\right)\left(a+c\right)\\b^2+1=b^2+ab+bc+ca=\left(b+c\right)\left(b+a\right)\\c^2+1=c^2+ab+bc+ca=\left(c+a\right)\left(c+b\right)\end{matrix}\right.\)

Thay vào biểu thức ta được :

\(\sqrt{\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)}=\sqrt{\left(a+b\right)\left(a+c\right)\left(b+c\right)\left(b+a\right)\left(c+a\right)\left(c+b\right)}=\sqrt{\left[\left(a+b\right)\left(b+c\right)\left(c+a\right)\right]^2}=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

Vậy biểu thức trên là một số hữu tỉ .

Wish you study well !!!

còn ai bik làm bài 1 , bài 2 nữa ko giúp mik vs

ngảy mai là đi học r

thanks

Link câu 2:Câu hỏi của nguyen thi thu hien - Toán lớp 9 | Học trực tuyến

Đặt \(\left\{{}\begin{matrix}a-b=x\\b-c=y\\c-a=z\end{matrix}\right.\Leftrightarrow x+y+z=0\)

\(\Leftrightarrow A=\sqrt{\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}}\\ \Leftrightarrow A=\sqrt{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2-2\left(\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{xz}\right)}\\ \Leftrightarrow A=\sqrt{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2-\dfrac{2\left(x+y+z\right)}{xyz}}\\ \Leftrightarrow A=\sqrt{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2-\dfrac{2\cdot0}{xyz}}\\ \Leftrightarrow A=\sqrt{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2}=\left|\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right|\left(đpcm\right)\)

Lời giải:
\(\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(a-c)^2}=(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a})^2-2(\frac{1}{(a-b)(b-c)}+\frac{1}{(b-c)(c-a)}+\frac{1}{(a-b)(c-a)})\)

\(=(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a})^2-2.\frac{c-a+a-b+b-c}{(a-b)(b-c)(c-a)}=(\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a})^2\)

\(\Rightarrow \sqrt{\frac{1}{(a-b)^2}+\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}}=|\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}|\) là số hữu tỷ (đpcm)

• Vì a, b, c đều dương và a + b + c = 2

nên \(0< a,b,c< 2\)

• Theo gt, ta có:

\(\Leftrightarrow\left\{{}\begin{matrix}b+c=2-a\\\left(b+c\right)^2-2bc=2-a^2\end{matrix}\right.\)

\(\Rightarrow\left(2-a\right)^2-2+a^2=2bc\)

\(\Rightarrow bc=\dfrac{\left(4-4a+a^2\right)-2+a^2}{2}=\dfrac{2a^2-4a+2}{2}=\left(a-1\right)^2\)

\(\Rightarrow b^2c^2=\left(a-1\right)^4\)

• Ta lại có: \(a\sqrt{\dfrac{\left(1+b^2\right)\left(1+c^2\right)}{1+a^2}}=a\sqrt{\dfrac{1+b^2+c^2+b^2c^2}{1+a^2}}\)

\(=a\sqrt{\dfrac{3-a^2+\left(a-1\right)^4}{1+a^2}}=a\sqrt{\dfrac{a^4-4a^3+5a^2-4a-4}{1+a^2}}\)

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

• Tương tự, ta cũng có: \(b\sqrt{\dfrac{\left(1+a^2\right)\left(1+c^2\right)}{1+b^2}}=b\left(2-b\right)\)

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

• Suy ra \(a\sqrt{\dfrac{\left(1+a^2\right)\left(a-2\right)^2}{1+a^2}}+b\sqrt{\dfrac{\left(1+a^2\right)\left(1+c^2\right)}{1+b^2}}+c\sqrt{\dfrac{\left(1+b^2\right)\left(1+a^2\right)}{1+c^2}}\)

\(=2\left(a+b+c\right)-\left(a^2+b^2+c^2\right)=2\left(đpcm\right)\)

\(\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2=a+b+c+2\left(\sqrt{ab}+\sqrt{ac}+\sqrt{bc}\right)=4\)

\(\Leftrightarrow\sqrt{ab}+\sqrt{ac}+\sqrt{bc}=1\)

\(\Rightarrow a+1=a+\sqrt{ab}+\sqrt{ac}+\sqrt{bc}=\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{c}\right)\)

Tương tự: \(b+1=\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{b}+\sqrt{c}\right)\)

\(c+1=\left(\sqrt{a}+\sqrt{c}\right)\left(\sqrt{b}+\sqrt{c}\right)\)

\(VT=\sum\dfrac{\sqrt{a}}{a+1}=\sum\dfrac{\sqrt{a}}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{c}\right)}\)

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

\(=\dfrac{2\left(\sqrt{ab}+\sqrt{ac}+\sqrt{bc}\right)}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{c}\right)\left(\sqrt{b}+\sqrt{c}\right)}=\dfrac{2}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{c}\right)\left(\sqrt{b}+\sqrt{c}\right)}\)

\(VP=\dfrac{2}{\sqrt{\left(a+1\right)\left(b+1\right)\left(c+1\right)}}=\dfrac{2}{\sqrt{\left(\sqrt{a}+\sqrt{b}\right)^2\left(\sqrt{a}+\sqrt{c}\right)^2\left(\sqrt{b}+\sqrt{c}\right)^2}}\)

\(=\dfrac{2}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}+\sqrt{c}\right)\left(\sqrt{b}+\sqrt{c}\right)}\)

\(\Rightarrow VT=VP\) (đpcm)