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\(\sqrt{ab}-\sqrt{a}-\sqrt{b}+1\)
\(=\sqrt{a}\left(\sqrt{b}-1\right)-\left(\sqrt{b}-1\right)\)
\(=\left(\sqrt{a}-1\right)\left(\sqrt{b}-1\right)\)
\(\sqrt{ab}-\sqrt{a}-\sqrt{b}+1=\sqrt{a}\left(\sqrt{b}-1\right)-\left(\sqrt{b}-1\right)=\left(\sqrt{a}-1\right)\left(\sqrt{b}-1\right)\)
\(=\sqrt{a}\left(\sqrt{a}+1\right)+2\sqrt{b}\left(\sqrt{a}+1\right)=\left(\sqrt{a}+1\right)\left(\sqrt{a}+2\sqrt{b}\right)\)
a, \(1-a\sqrt{a}\)
\(=\left[1-\left(\sqrt{a}\right)^3\right]\)
\(=\left(1-\sqrt{a}\right)\left[\left(\sqrt{a}\right)^2+1.\sqrt{a}+1^2\right]\)
\(=\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)\)
b, \(x-2\sqrt{x-1}\)
\(=\left(x-1\right)-2\sqrt{x-1}+1\)
\(=\left[\left(\sqrt{x-1}\right)-1\right]^2\)
Lời giải :
\(\sqrt{a-b}-\sqrt{a^2-b^2}\)
\(=\sqrt{a-b}-\sqrt{a-b}\cdot\sqrt{a+b}\)
\(=\sqrt{a-b}\left(1-\sqrt{a+b}\right)\)
\(x-y=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)\)
\(a\sqrt{b}+b\sqrt{a}=\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)\)
\(\sqrt{21}+\sqrt{3}+\sqrt{7}+1\)
\(=\sqrt{3}\left(\sqrt{7}+1\right)+\left(\sqrt{7}+1\right)\)
\(=\left(\sqrt{7}+1\right)\left(\sqrt{3}+1\right)\)
\(\sqrt{1-a}+\sqrt{1-a^2}\)
\(=\sqrt{1-a}+\sqrt{\left(1-a\right)\left(1+a\right)}\)
\(=\sqrt{1-a}\left(1+\sqrt{1+a}\right)\)
\(ab+b\sqrt{a}+\sqrt{a}+1\)
(đk: \(a\ge0\))
\(=b\sqrt{a}\left(\sqrt{a}+1\right)+\sqrt{a}+1=\left(\sqrt{a}+1\right)\left(b\sqrt{a}+1\right)\)
ĐK: \(x,y\ge0\)
\(\sqrt{x^3}-\sqrt{y^3}+\sqrt{x^2y}-\sqrt{xy^2}=x\left(\sqrt{x}+\sqrt{y}\right)-y\left(\sqrt{x}+\sqrt{y}\right)=\left(\sqrt{x}+\sqrt{y}\right)\left(x-y\right)\)
\(=\left(\sqrt{x}+\sqrt{y}\right)^2\left(\sqrt{x}-\sqrt{y}\right)\)
A)=a+\(2\sqrt{a}+2\sqrt{a}\)+4
=\(\sqrt{a}\left(\sqrt{a}+2\right)+2\left(\sqrt{a}+2\right)=\left(\sqrt{a}+2\right)^2\)
b)= \(\left(a-\sqrt{7}\right)\left(a+\sqrt{7}\right)\)
c) \(\sqrt{a}\left(\sqrt{b}-4\right)+3\cdot\left(\sqrt{b}-4\right)=\left(\sqrt{a}+3\right)\left(\sqrt{b}-4\right)\)
\(a\sqrt{a}+2a+\sqrt{a}+2=\left(a\sqrt{a}+2a\right)+\left(\sqrt{a}+2\right)\)
\(=a\left(\sqrt{a}+2\right)+\left(\sqrt{a}+2\right)=\left(\sqrt{a}+2\right)\left(a+1\right)\)
\(a\sqrt{a}+2a+\sqrt{a}+2=a\left(\sqrt{a}+2\right)+\left(\sqrt{a}+2\right)=\left(a+1\right)\left(\sqrt{a}+2\right)\)