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![](https://rs.olm.vn/images/avt/0.png?1311)
\(\sqrt{ab}=\sqrt{-a}.\sqrt{-b}\) (vì a<0 , b<0)
Áp dụng : \(\sqrt{\left(-25\right).\left(-64\right)}=\sqrt{-\left(-25\right)}.\sqrt{-\left(-64\right)}=\sqrt{25}.\sqrt{64}=5.8=40\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a: \(=\sqrt{3a}:\sqrt{b}\)
b: \(=\sqrt{a}:\sqrt{xy}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\sqrt{\dfrac{a}{b}}\)=\(\dfrac{\sqrt{a}}{\sqrt{b}}\) với a,b<0
Ta có : \(\sqrt{\dfrac{-49}{-81}}\)=\(\sqrt{\dfrac{49}{81}}\)=\(\dfrac{7}{9}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Lời giải:
\(a+b+c=\sqrt{a}+\sqrt{b}+\sqrt{c}=2\)
\(\Rightarrow (\sqrt{a}+\sqrt{b}+\sqrt{c})^2=4\)
\(\Leftrightarrow a+b+c+2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})=4\)
\(\Leftrightarrow \sqrt{ab}+\sqrt{bc}+\sqrt{ac}=\frac{4-(a+b+c)}{2}=1\)
\(\Rightarrow a+1=a+\sqrt{ab}+\sqrt{bc}+\sqrt{ac}=(\sqrt{a}+\sqrt{b})(\sqrt{a}+\sqrt{c})\)
Tương tự:
$b+1=(\sqrt{b}+\sqrt{c})(\sqrt{c}+\sqrt{a})$
$c+1=(\sqrt{c}+\sqrt{a})(\sqrt{c}+\sqrt{b})$
Khi đó:
\(A=\left[\frac{\sqrt{a}}{(\sqrt{a}+\sqrt{b})(\sqrt{a}+\sqrt{c})}+\frac{\sqrt{b}}{(\sqrt{b}+\sqrt{a})(\sqrt{b}+\sqrt{c})}+\frac{\sqrt{c}}{(\sqrt{c}+\sqrt{a})(\sqrt{c}+\sqrt{b})}\right]\sqrt{(a+1)(b+1)(c+1)}\)
\(\frac{\sqrt{a}(\sqrt{b}+\sqrt{c})+\sqrt{b}(\sqrt{c}+\sqrt{a})+\sqrt{c}(\sqrt{a}+\sqrt{b})}{(\sqrt{a}+\sqrt{b})(\sqrt{b}+\sqrt{c})(\sqrt{c}+\sqrt{a})}.\sqrt{(\sqrt{a}+\sqrt{b})^2(\sqrt{b}+\sqrt{c})^2(\sqrt{c}+\sqrt{a})^2}\)
\(=\frac{2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})}{(\sqrt{a}+\sqrt{b})(\sqrt{b}+\sqrt{c})(\sqrt{c}+\sqrt{a})}.(\sqrt{a}+\sqrt{b})(\sqrt{b}+\sqrt{c})(\sqrt{c}+\sqrt{a})\)
\(=2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})=2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Với \(\begin{cases}a< 0\\b< 0\end{cases}\) thì \(\sqrt{\frac{a}{b}}=\sqrt{-a}:\sqrt{-b}\)
Áp dụng \(\sqrt{\frac{-49}{-81}}=\sqrt{-\left(-49\right)}:\sqrt{-\left(-81\right)}=\sqrt{49}:\sqrt{81}=7:9=\frac{7}{9}\)
Do a và b âm nên -a và -b dương
Khi đó , ta có: \(\sqrt{a.b}=\sqrt{\left(-a\right)\left(-b\right)}=\sqrt{-a}.\sqrt{-b}\)
Áp dụng , ta có: \(\sqrt{\left(-25\right)\left(-64\right)}=\sqrt{25}.\sqrt{64}=5.8=40\)