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

\(\frac{a+\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=\frac{\left(a+\sqrt{ab}\right)\left(\sqrt{a}-\sqrt{b}\right)}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}=\frac{\left(a+\sqrt{ab}\right)\left(\sqrt{a}-\sqrt{b}\right)}{a-b}\)

a, = \(\sqrt{a^2b^2.\left(1+\frac{1}{a^2b^2}\right)}\) = \(\sqrt{a^2b^2+1}\)

c, = \(\sqrt{\frac{a+ab}{b^4}}\) = \(\frac{\sqrt{a+ab}}{b^2}\)

k mk nha

a, \(ab\sqrt{1+\frac{1}{a^2b^2}}\)

 \(ab\sqrt{1+\frac{1}{a^2b^2}}=ab\sqrt{\frac{1+a^2b^2}{a^2b^2}}=\frac{ab}{\left|ab\right|}\sqrt{1+a^2b^2}\)

\(=\hept{\begin{cases}\sqrt{1+a^2b^2}ĐK:ab>0\\-\sqrt{1+a^2b^2}ĐKab< 0\end{cases}}\)

b, \(\sqrt{\frac{a}{b^3}+\frac{a}{b^4}}\)

\(\sqrt{\frac{a}{b^3}+\frac{a}{b^4}}=\sqrt{\frac{a+ab}{b^4}}=\frac{1}{b^2}\sqrt{a+ab}\)

\(\frac{a+\sqrt{ab}}{b+\sqrt{ab}}=\frac{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)}=\frac{\sqrt{ab}}{b}\)

a/ \(\sqrt{8\left(\sqrt{2}-\sqrt{3}\right)^2}=2\sqrt{2}\left(\sqrt{3}-\sqrt{2}\right)=2\sqrt{6}-4\)

b/ \(ab\sqrt{1+\frac{1}{a^2b^2}}=ab.\sqrt{\frac{a^2b^2+1}{a^2b^2}}=\sqrt{a^2b^2.\frac{a^2b^2+1}{a^2b^2}}=\sqrt{a^2b^2+1}\)

c/ \(\sqrt{\frac{a}{b^3}+\frac{a}{b^4}}=\sqrt{\frac{a}{b^3}\left(1+\frac{1}{b}\right)}=\frac{1}{b}.\sqrt{\frac{a}{b}\left(1+\frac{1}{b}\right)}\)

d/ \(\frac{a+\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=\frac{\sqrt{a}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}+\sqrt{b}}=\sqrt{a}\)

Lời giải:

\(\sqrt{\frac{(1+\sqrt{2})^3}{27}}=\sqrt{\frac{(1+\sqrt{2})^3}{3^3}}=\sqrt{\frac{3(1+\sqrt{2})^3}{3^4}}\)

\(=\frac{(1+\sqrt{2})\sqrt{3+3\sqrt{2}}}{9}\)

\(ab\sqrt{\frac{1}{a}+\frac{1}{b}}=\sqrt{(ab)^2(\frac{1}{a}+\frac{1}{b})}=\sqrt{ab^2+a^2b}\)

\(\sqrt{\frac{1}{600}}=\sqrt{\frac{6}{3600}}=\frac{\sqrt{6}}{\sqrt{3600}}=\frac{\sqrt{6}}{60}\)

\(\sqrt{\frac{11}{540}}=\sqrt{\frac{11}{36.15}}=\frac{1}{6}\sqrt{\frac{165}{15^2}}=\frac{1}{6}.\frac{\sqrt{165}}{15}=\frac{\sqrt{165}}{90}\)

\(\sqrt{\frac{3}{50}}=\sqrt{\frac{3}{25.2}}=\frac{1}{5}\sqrt{\frac{3}{2}}=\frac{1}{5}\sqrt{\frac{6}{4}}=\frac{1}{5}.\frac{\sqrt{6}}{2}=\frac{\sqrt{6}}{10}\)

\(\sqrt{\frac{5}{98}}=\sqrt{\frac{5}{49.2}}=\frac{1}{7}\sqrt{\frac{5}{2}}=\frac{1}{7}.\sqrt{\frac{10}{4}}=\frac{\sqrt{10}}{14}\)

\(\sqrt{\frac{\left(1-\sqrt{3}\right)^2}{27}}=\frac{\left|1-\sqrt{3}\right|}{\sqrt{9.3}}=\frac{\sqrt{3}-1}{3\sqrt{3}}=\frac{\sqrt{3}\left(\sqrt{3}-1\right)}{9}\)