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a, \(5+\sqrt{5}=\sqrt{5}\left(\sqrt{5}+1\right)\)
b, \(a-2\sqrt{a}=\sqrt{a}\left(\sqrt{a}-2\right)\)
c, \(x-\sqrt{xy}=\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)\)
d, \(x-y-\sqrt{x}-\sqrt{y}\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)-\left(\sqrt{x}+\sqrt{y}\right)\)
\(=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}-1\right)\)
3/a) \(BĐT\Leftrightarrow\left(\sqrt{x}-\sqrt{y}\right)^2\ge0\)(đúng với mọi x, y không âm)
Đẳng thức xảy ra khi x = y
b) \(BĐT\Leftrightarrow\frac{\left(x-y\right)^2}{xy}\ge0\) (đúng với mọi x, y không âm)
"=" <=> x = y
c) BĐT \(\Leftrightarrow2a+2b+2\ge2\sqrt{ab}+2\sqrt{a}+2\sqrt{b}\)
\(\Leftrightarrow\left(a-2\sqrt{ab}+b\right)+\left(a-2\sqrt{a}+1\right)+\left(b-2\sqrt{b}+1\right)\ge0\)
\(\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2+\left(\sqrt{a}-1\right)^2+\left(\sqrt{b}-1\right)^2\ge0\) (đúng)
"=" <=> a = b = 1
1/ \(A=\sqrt{7-2\sqrt{7}.1+1}-\sqrt{7-2\sqrt{7}.\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}-\sqrt{2}\right)^2}\)
\(=\left|\sqrt{7}-1\right|-\left|\sqrt{7}-\sqrt{2}\right|\) (thực ra em nghĩ ko cần thêm trị tuyệt đối đâu nhưng thêm cho chắc:D)
\(=\sqrt{7}-1-\sqrt{7}+\sqrt{2}=\sqrt{2}-1\)
2/Em thấy nó sai sai nên thôi:(
a) \(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}\)
\(=a\sqrt{a}-b\sqrt{b}+a\sqrt{b}-b\sqrt{a}\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)-\left(\sqrt{a}-\sqrt{b}\right)\sqrt{ab}\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b-\sqrt{ab}\right)\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(a+b\right)\)
b) \(x-y+\sqrt{xy^2}-\sqrt{y^3}\)
\(=\left(x-y\right)+\left(y\sqrt{x}-y\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)+y\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}+y\right)\)
a, \(7\sqrt{AB}+7B-\sqrt{A}-\sqrt{B}=7\sqrt{B}\left(\sqrt{A}+\sqrt{B}\right)-\left(\sqrt{A}+\sqrt{B}\right)\)\(=\left(\sqrt{A}+\sqrt{B}\right)\left(7\sqrt{B}-1\right)\)
b, \(a\sqrt{b}-b\sqrt{a}+\sqrt{a}-\sqrt{b}=\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)+\left(\sqrt{a}-\sqrt{b}\right)\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{ab}+1\right)\)
c,\(\sqrt{x^2-25y^2}-\sqrt{x-5y}=\sqrt{x-5y}.\sqrt{x+5y}-\sqrt{x-5y}\)
\(=\sqrt{x-5y}\left(\sqrt{x+5y}-1\right)\)
\(a,7\sqrt{AB}+7B-\sqrt{A}-\sqrt{B}\)( Với A>= 0, B>=0)
\(=\left(7\sqrt{AB}-\sqrt{A}\right)+\left(7B-\sqrt{B}\right)\)
\(=7\sqrt{A}\left(\sqrt{B}-1\right)+7\sqrt{B}\left(\sqrt{B}-1\right)\)
\(=\left(\sqrt{B}-1\right)\left(7\sqrt{A}+7\sqrt{B}\right)\)
\(=7\left(\sqrt{B}-1\right)\left(\sqrt{A}+\sqrt{B}\right)\)
\(b,a\sqrt{b}-b\sqrt{a}+\sqrt{a}-\sqrt{b}\)Với a>= 0, b>=0)
\(=\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)+\left(\sqrt{a}-\sqrt{b}\right)\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{ab}+1\right)\)
\(c,\sqrt{x^2-25y^2}-\sqrt{x-5y}\)
\(=\sqrt{\left(x-5y\right)\left(x+5y\right)}-\sqrt{x-5y}\)
\(=\sqrt{x-5y}.\sqrt{x+5y}-\sqrt{x-5y}\)
\(=\sqrt{x-5y}\left(\sqrt{x+5y}-1\right)\)
\(a,\)\(7\sqrt{ab}+7b-\sqrt{a}-\sqrt{b}\)
\(=7\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)-\left(\sqrt{a}+\sqrt{b}\right)\)
\(=\left(\sqrt{a}+\sqrt{b}\right)\left(7\sqrt{b}-1\right)\)
\(b,a\sqrt{b}-b\sqrt{a}+\sqrt{a}-\sqrt{b}\)
\(=\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)+\left(\sqrt{a}-\sqrt{b}\right)\)
\(=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{ab}-1\right)\)
\(c,\sqrt{x^2-25y^2}-\sqrt{x-5y}\)
\(=\sqrt{\left(x-5y\right)\left(x+5y\right)}-\sqrt{x-5y}\)
\(=\sqrt{x-5y}\left(\sqrt{x-5y}-1\right)\)
a)\(a-5\sqrt{a}=\sqrt{a}\left(\sqrt{a}-5\right)\)
b)\(a-7=\left(\sqrt{a}-\sqrt{7}\right)\left(\sqrt{a}+\sqrt{7}\right)\)
c)\(a+4\sqrt{a}+4=\left(\sqrt{a}+2\right)^2\)
d)\(\sqrt{xy}-4\sqrt{x}+3\sqrt{y}-12=\sqrt{x}\left(\sqrt{y}-4\right)+3\left(\sqrt{y}-4\right)=\left(\sqrt{x}+3\right)\left(\sqrt{y}-4\right)\)
em cảm ơn ạ