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a) \(\frac{2\sqrt{5}+b\sqrt{a}}{\sqrt{ab}}-\frac{1}{\sqrt{a}-\sqrt{b}}\)
\(=\frac{2\sqrt{5ab}+ab\sqrt{b}}{ab}-\frac{\sqrt{a}+\sqrt{b}}{a-b}\)
\(=\frac{\left(2\sqrt{5ab}+ab\sqrt{b}\right)\left(a-b\right)-\left(\sqrt{a}+\sqrt{b}\right)ab}{ab\left(a-b\right)}\)
\(=\frac{2a\sqrt{5ab}-2b\sqrt{5ab}+a^2b\sqrt{b}-ab^2\sqrt{b}-ab\sqrt{a}-ab\sqrt{b}}{ab\left(a-b\right)}\)
b) \(\left(1+\frac{a-\sqrt{a}}{\sqrt{a}-1}\right)\cdot\left(\frac{1+1+\sqrt{a}}{1-a^2}\right)^2\)
\(=\left(1+\frac{\left(\sqrt{a}-1\right)\sqrt{a}}{\sqrt{a}-1}\right)\cdot\left(\frac{2+\sqrt{a}}{\left(1-a\right)\left(1+a\right)}\right)^2\)
\(=\left(1+\sqrt{a}\right)\left[\frac{2+\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)\left(1+a\right)}\right]^2\)
\(=\frac{4+4\sqrt{a}+a}{\left(1-\sqrt{a}\right)^2\left(1+\sqrt{a}\right)\left(1+a\right)^2}\) \(a\ge0;a\ne\pm1\)
Bổ sung đk của phần a: \(a,b>0;a\ne b\)
A= \(\left(\frac{\sqrt{b}}{a-\sqrt{ab}}-\frac{\sqrt{a}}{\sqrt{ab}-b}\right).\left(a\sqrt{b}-b\sqrt{a}\right)\)
A = \(\left(\frac{\sqrt{b}}{\sqrt{a}.\sqrt{a}-\sqrt{ab}}-\frac{\sqrt{a}}{\sqrt{ab}-\sqrt{b}.\sqrt{b}}\right).\left(a\sqrt{b}-b\sqrt{a}\right)\)
A = \(\left(\frac{\sqrt{b}}{\sqrt{a}\left(\sqrt{a}-\sqrt{b}\right)}-\frac{\sqrt{a}}{\sqrt{b}\left(\sqrt{a}-\sqrt{b}\right)}\right).\left(a\sqrt{b}-b\sqrt{a}\right)\)
A = \(\left(\frac{b}{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}-\frac{a}{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}\right)\left(a\sqrt{b}-b\sqrt{a}\right)\)
A = \(\left(\frac{b-a}{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}\right).\left(\sqrt{a}.\sqrt{a}.\sqrt{b}-\sqrt{b}.\sqrt{b}\sqrt{a}\right)\)
A = \(\left(\frac{b-a}{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}\right).\left(\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)\right)\)
A = b-a
B = \(\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{\sqrt{a}}{a-\sqrt{a}}\right):\frac{\sqrt{a}+1}{a-1}\)
B = \(\left(\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{a-1}-\frac{\sqrt{a}\left(a+\sqrt{a}\right)}{a^2-a}\right).\frac{a-1}{\sqrt{a}+1}\)
B = \(\left(\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{a-1}-\frac{\sqrt{a}.\sqrt{a}\left(\sqrt{a}+1\right)}{a\left(a-1\right)}\right).\frac{a-1}{\sqrt{a}+1}\)
\(B=\left(\frac{a\sqrt{a}\left(\sqrt{a}+1\right)}{a\left(a-1\right)}-\frac{a\left(\sqrt{a}+1\right)}{a\left(a-1\right)}\right).\frac{a-1}{\sqrt{a}+1}\)
B= \(\left(\frac{a\sqrt{a}\left(\sqrt{a}+1\right)-a\left(\sqrt{a}+1\right)}{a\left(a-1\right)}\right).\frac{a-1}{\sqrt{a}+1}\)
B= \(\left(\frac{\left(\sqrt{a}+1\right)\left(a\sqrt{a}-a\right)}{a\left(a-1\right)}\right).\frac{a-1}{\sqrt{a}+1}\)
B = \(\frac{\left(\sqrt{a}+1\right)a\left(\sqrt{a}-1\right)}{a\left(a-1\right)}.\frac{a-1}{\sqrt{a}+1}\)
\(B=\frac{a\left(\sqrt{a}^2-1^2\right)}{a\left(a-1\right)}.\frac{a-1}{\sqrt{a}+1}\)
\(B=\frac{a\left(a-1\right)}{a\left(a-1\right)}.\frac{a-1}{\sqrt{a}+1}\)
B = \(\frac{a-1}{\sqrt{a}+1}\)
a) Ta có: \(\frac{a-b}{\sqrt{a}-\sqrt{b}}-\frac{\sqrt{a^3}-\sqrt{b^3}}{a-b}\)
\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}-\sqrt{b}}-\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)}{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}\)
\(=\frac{\left(\sqrt{a}+\sqrt{b}\right)^2}{\sqrt{a}+\sqrt{b}}-\frac{a+\sqrt{ab}+b}{\sqrt{a}+\sqrt{b}}\)
\(=\frac{a+2\sqrt{ab}+b-a-\sqrt{ab}-b}{\sqrt{a}+\sqrt{b}}\)
\(=\frac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}\)
b)Sửa đề: \(\frac{\left(\sqrt{a}+\sqrt{b}\right)^2-4\sqrt{ab}}{\sqrt{a}-\sqrt{b}}-\frac{a\sqrt{b}+b\sqrt{a}}{\sqrt{ab}}\)
Ta có: \(\frac{\left(\sqrt{a}+\sqrt{b}\right)^2-4\sqrt{ab}}{\sqrt{a}-\sqrt{b}}-\frac{a\sqrt{b}+b\sqrt{a}}{\sqrt{ab}}\)
\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\left(\sqrt{a}-\sqrt{b}\right)}-\frac{\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{ab}}\)
\(=\sqrt{a}-\sqrt{b}-\sqrt{a}-\sqrt{b}\)
\(=-2\sqrt{b}\)
c) Ta có: \(\left(\frac{1}{\sqrt{a}-1}-\frac{1}{\sqrt{a}}\right):\left(\frac{\sqrt{a}+1}{\sqrt{a}-2}-\frac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)
\(=\left(\frac{\sqrt{a}}{\sqrt{a}\left(\sqrt{a}-1\right)}-\frac{\sqrt{a}-1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\left(\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}-\frac{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\right)\)
\(=\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{a-1-a+4}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\)
\(=\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\frac{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}{3}\)
\(=\frac{\sqrt{a}-2}{3\sqrt{a}}\)
d) Ta có: \(\left(\frac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}\right)\left(\frac{\sqrt{a}+\sqrt{b}}{a-b}\right)^2\)
\(=\left(\frac{\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)}{\left(\sqrt{a}+\sqrt{b}\right)}-\sqrt{ab}\right)\left(\frac{\left(\sqrt{a}+\sqrt{b}\right)}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}\right)^2\)
\(=\left(a-\sqrt{ab}+b-\sqrt{ab}\right)\cdot\left(\frac{1}{\sqrt{a}-\sqrt{b}}\right)^2\)
\(=\left(a-2\sqrt{ab}+b\right)\cdot\frac{1}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\left(\sqrt{a}-\sqrt{b}\right)^2}=1\)
e) Ta có: \(\left(\frac{\sqrt{x}}{3+\sqrt{x}}+\frac{x+9}{9-x}\right):\left(\frac{3\sqrt{x}+1}{x-3\sqrt{x}}-\frac{1}{\sqrt{x}}\right)\)
\(=\left(\frac{\sqrt{x}\left(3-\sqrt{x}\right)}{\left(3+\sqrt{x}\right)\left(3-\sqrt{x}\right)}+\frac{x+9}{\left(3+\sqrt{x}\right)\left(3-\sqrt{x}\right)}\right):\left(\frac{3\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-3\right)}-\frac{\sqrt{x}-3}{\sqrt{x}\left(\sqrt{x}-3\right)}\right)\)
\(=\frac{3\sqrt{x}+9}{\left(3+\sqrt{x}\right)\left(3-\sqrt{x}\right)}:\frac{3\sqrt{x}+1-\sqrt{x}+3}{\sqrt{x}\left(\sqrt{x}-3\right)}\)
\(=\frac{3\left(\sqrt{x}+3\right)}{-\left(\sqrt{x}-3\right)\cdot\left(\sqrt{x}+3\right)}\cdot\frac{\sqrt{x}\left(\sqrt{x}-3\right)}{2\left(\sqrt{x}+2\right)}\)
\(=\frac{-3\sqrt{x}}{2\sqrt{x}+4}\)
1/ a/ \(\sqrt{0,9.0,16.0,4}=\sqrt{\frac{9.16.4}{10000}}=\sqrt{\frac{\left(3.4.2\right)^2}{10^4}}=\frac{24}{1010}=\frac{6}{25}\)
b/ \(\sqrt{0,0016}=\sqrt{\frac{16}{100}}=\frac{4}{10}=\frac{2}{5}\)
c/ \(\frac{\sqrt{72}}{\sqrt{2}}=\frac{\sqrt{2}.\sqrt{36}}{\sqrt{2}}=\sqrt{36}=6\)
d/ \(\frac{\sqrt{2}}{\sqrt{288}}=\frac{\sqrt{2}}{\sqrt{2}.\sqrt{144}}=\frac{1}{\sqrt{144}}=\frac{1}{12}\)
2.
a/ \(\frac{2}{a}.\sqrt{\frac{16a^2}{9}}=\frac{2}{a}.\frac{4\left|a\right|}{3}=-\frac{8a}{3a}=-\frac{8}{3}\) (Vì a<0)
b/ \(\frac{3}{a-1}.\sqrt{\frac{4a^2-8a+4}{25}}=\frac{3}{a-1}.\sqrt{\frac{4\left(a-1\right)^2}{25}}=\frac{3.2\left|a-1\right|}{5.\left(a-1\right)}=\frac{6\left(a-1\right)}{5\left(a-1\right)}=\frac{6}{5}\)
c/ \(\frac{\sqrt{243a}}{\sqrt{3a}}=\frac{9\sqrt{3a}}{\sqrt{3a}}=9\)
d/ \(\frac{3\sqrt{18a^2b^4}}{\sqrt{2a^2b^2}}=\frac{3.3\sqrt{2}.\left|a\right|.\left|b\right|^2}{\sqrt{2}.\left|a\right|.\left|b\right|}=9\left|b\right|\)
a/ \(\frac{2}{a}.\frac{4\left|a\right|}{3}=\frac{-8a}{3a}=-\frac{8}{3}\)
b/ \(\frac{3}{a-1}\sqrt{\frac{4\left(a-1\right)^2}{25}}=\frac{3}{\left(a-1\right)}.\frac{2\left|a-1\right|}{5}=\frac{6\left(a-1\right)}{5\left(a-1\right)}=\frac{6}{5}\)
c/ \(\frac{3\sqrt{9a^2b^4}}{\sqrt{a^2b^2}}=\frac{9.\left|a\right|.b^2}{\left|a\right|\left|b\right|}=9\left|b\right|\)
d/ \(\left(1+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\left(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)
a/ \(=\frac{2}{a}.\frac{4\left|a\right|}{3}=\frac{2}{a}.\frac{-4a}{3}=\frac{-8}{3}\)
b/ \(=\frac{3}{a-1}.\frac{\left|2a-2\right|}{5}=\frac{3}{a-1}.\frac{2\left(a-1\right)}{5}=\frac{6}{5}\)
c/ \(=\sqrt{\frac{162a^2b^4}{2a^2b^2}}=\sqrt{81b^2}=9\left|b\right|\)
d/ \(=\left(1+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\left(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)\)
\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)