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\(\frac{1}{7}\sqrt{51}< \frac{1}{7}\sqrt{64}=\frac{1}{7}.8=\frac{8}{7}\)
\(\frac{1}{9}\sqrt{150}>\frac{1}{9}\sqrt{144}=\frac{1}{9}.12=\frac{4}{3}\)
\(1+\frac{1}{3}>1+\frac{1}{7}\Rightarrow\frac{4}{3}>\frac{8}{7}\)
Do đó: \(\frac{1}{7}\sqrt{51}< \frac{1}{9}\sqrt{150}\)
a)1/7\(\sqrt{51}\)=\(\sqrt{\frac{51}{49}}\);1/9\(\sqrt{150}=\sqrt{\frac{150}{81}}=\sqrt{\frac{50}{27}}\)
\(\frac{51}{49}=1+\frac{1}{49}+\frac{1}{49}\);\(\frac{50}{27}=1+\frac{23}{27}>1+\frac{23}{36}>\)\(1+\frac{2}{36}=1+\frac{1}{36}+\frac{1}{36}\)
1/49<1/36 nên 51/49<50/27 =>1/7\(\sqrt{51}\)<1/9\(\sqrt{150}\)
b) \(\sqrt{2017}+\sqrt{2016}>\sqrt{2016}\)+\(\sqrt{2015}\)
=>\(\frac{1}{\sqrt{2017}+\sqrt{2016}}< \)\(\frac{1}{\sqrt{2016}+\sqrt{ }2015}\) <=> \(\sqrt{2017}-\sqrt{2016}< \sqrt{2016}\)-\(\sqrt{2015}\)
\(\frac{1}{7}\)\(\sqrt{51}\)=\(\frac{\sqrt{51}}{7}\)=\(\sqrt{\frac{51}{49}}\)=\(\sqrt{\frac{4131}{3969}}\)
\(\frac{1}{9}\)\(\sqrt{150}\)=\(\frac{\sqrt{150}}{9}\)=\(\sqrt{\frac{150}{81}}\)=\(\sqrt{\frac{7350}{3969}}\)
Mà \(\sqrt{\frac{4131}{3969}}\)<\(\sqrt{\frac{7350}{3969}}\) ( Vì 4131<7350 )
⇒ĐPCM
a) Ta có: \(\frac{1}{5}\sqrt{150}=\frac{1}{5}\cdot5\sqrt{6}=\sqrt{6}=\frac{1}{3}\cdot\sqrt{6\cdot9}=\frac{1}{3}\sqrt{54}>\frac{1}{3}\sqrt{51}\)
b) Ta có: \(\frac{1}{2}\sqrt{6}=\sqrt{\frac{6}{4}}< \sqrt{\frac{36}{2}}=6\sqrt{\frac{1}{2}}\)
a) Vì \(5,\left(6\right)< 6\)\(\Rightarrow\)\(\frac{51}{9}< \frac{150}{25}\)
\(\Rightarrow\)\(\sqrt{\frac{51}{9}}< \sqrt{\frac{150}{25}}\)
\(\Rightarrow\)\(\frac{1}{3}\sqrt{51}< \frac{1}{5}\sqrt{150}\)
b) Vì \(1,5< 18\)\(\Rightarrow\)\(\frac{6}{4}< \frac{36}{2}\)
\(\Rightarrow\)\(\sqrt{\frac{6}{4}}< \sqrt{\frac{36}{2}}\)
\(\Rightarrow\)\(\frac{1}{2}\sqrt{6}< 6\sqrt{\frac{1}{2}}\)
\(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+.....+\frac{1}{\sqrt{100}}>\frac{1}{\sqrt{100}}+\frac{1}{\sqrt{100}}+....+\frac{1}{\sqrt{100}}\)
\(\Leftrightarrow\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+....+\frac{1}{\sqrt{100}}>100.\frac{1}{\sqrt{100}}=10.\)
\(\frac{\sqrt{51}}{7}< \frac{\sqrt{64}}{7}=\frac{8}{7}< \frac{4}{3}=\frac{\sqrt{144}}{9}< \frac{\sqrt{150}}{9}\)
Nên \(\frac{\sqrt{51}}{7}< \frac{\sqrt{150}}{9}\)