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a, \(\sqrt{27.98}=\sqrt{3^3.49.2}=21\sqrt{6}\)
b, \(\sqrt{125.96}=\sqrt{5^3.4^2.6}=20\sqrt{30}\)
c, \(\sqrt{500.162}=\sqrt{10^2.5.9^2.2}=90\sqrt{10}\)
d, \(\sqrt{a^4b^5}=a^2b^2\sqrt{b}\)
a) Ta có:
\(5\sqrt{a}-4b\sqrt{25a^3}+5a\sqrt{16ab^2}-2\sqrt{9a}\)
\(=5\sqrt{a}-4b.5a\sqrt{a}+5a.4b\sqrt{a}-2.3\sqrt{a}\)
\(=5\sqrt{a}-20ab\sqrt{a}+20ab\sqrt{a}-6\sqrt{a}\) \(=-\sqrt{a}\)
b) Ta có:
\(5a\sqrt{64ab^3}-\sqrt{3}.\sqrt{12a^3b^3}+2ab\sqrt{9ab}\) \(-5b\sqrt{81a^3b}\)
\(=5a.8b\sqrt{ab}-\sqrt{3.12a^3b^3}+2ab.3\sqrt{ab}\) \(-5b.9a\sqrt{ab}\)
\(=40ab\sqrt{ab}-6ab\sqrt{ab}+6ab\sqrt{ab}-45ab\)\(\sqrt{ab}\)
\(=-5ab\sqrt{ab}\)
a) \(\sqrt{\left(1-\sqrt{5}\right)^2}-\sqrt{\left(3-\sqrt{5}\right)^2}=\left(\sqrt{5}-1\right)-\left(3-\sqrt{5}\right)=2\sqrt{5}-4\)
b) \(\frac{a-2\sqrt{a}+1}{\sqrt{a}-1}=\frac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}-1}=\sqrt{a}-1\) ( \(a\ge0\ne1\))
c) \(\frac{a+\sqrt{a}}{a}=\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}.\sqrt{a}}=\frac{\sqrt{a}+1}{\sqrt{a}}=1+\frac{1}{\sqrt{a}}\)(\(a>0\))
d) \(\frac{3+\sqrt{3}}{1+\sqrt{3}}=\frac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}+1}=\sqrt{3}\)
b) \(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2+\sqrt{3}\right)^2}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}}\)\(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}}\)
\(=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}}\)\(=\sqrt{4+\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}}=\sqrt{4+\sqrt{5\sqrt{3}+25-5\sqrt{3}}}=\sqrt{4+5}=3\)
a) \(\sqrt{\left(5-\sqrt{3}\right)^2}=\left|5-\sqrt{3}\right|=5-\sqrt{3}\)
b) \(\sqrt{\left(1-\sqrt{2}\right)^2}=\left|1-\sqrt{2}\right|=-\left(1-\sqrt{2}\right)=\sqrt{2}-1\)( vì 1 < √2 )
c) \(\sqrt{\left(\sqrt{3}-2\right)^2}=\left|\sqrt{3}-2\right|=-\left(\sqrt{3}-2\right)=2-\sqrt{3}\)( vì √3 < 2 )