So sánh:
a) 4\(\sqrt{7}\) và 3\(\sqrt{13}\)
b) 3\(\sqrt{12}\) và 2\(\sqrt{16}\)
c) \(\frac{1}{4}\)\(\sqrt{82}\) và 6\(\sqrt{\frac{1}{7}}\)
d) \(\frac{1}{2}\)\(\sqrt{\frac{17}{2}}\) và \(\frac{1}{3}\)\(\sqrt{19}\)
e) 3\(\sqrt{3}\) -2\(\sqrt{2}\) và 2
f) \(\sqrt{7}\) + \(\sqrt{5}\) và \(\sqrt{49}\)
g) \(\sqrt{2}\) + \(\sqrt{11}\) và \(\sqrt{3}\) +5
h)\(\frac{1}{2}\) \(\sqrt{\frac{17}{2}}\) và \(\frac{1}{3}\) \(\sqrt{19}\)
i) \(\sqrt{21}\) -\(\sqrt{5}\) và \(\sqrt{20}\) -\(\sqrt{6}\)
j) \(\frac{1}{4}\) \(\sqrt{82}\) và 6\(\sqrt{\frac{1}{7}}\)
k) \(\sqrt{\sqrt{6}+\sqrt{20}}\) và \(\sqrt{1+\sqrt{5}}\)
l) \(\sqrt{7}\) -\(\sqrt{2}\) và 1
m) \(\sqrt{30}\) - \(\sqrt{29}\) và \(\sqrt{29}\)-\(\sqrt{28}\)
n) \(\sqrt{8}+\sqrt{5}\) và \(\sqrt{7}+\sqrt{6}\)
o) \(\sqrt{27}+\sqrt{6}+1\) và \(\sqrt{48}\)
p) 5\(\sqrt{2}\) + \(\sqrt{75}\) và 5\(\sqrt{3}\) +\(\sqrt{50}\)
q) \(\sqrt{5}\) - \(\sqrt{3}\) và \(\frac{1}{2}\)
a)
\(4\sqrt{7}=\sqrt{4^2.7}=\sqrt{112}\)
\(3\sqrt{13}=\sqrt{3^2.13}=\sqrt{117}\)
\(\sqrt{112}< \sqrt{117}\Rightarrow 4\sqrt{7}< 3\sqrt{13}\)
b) \(3\sqrt{12}=\sqrt{3^2.12}=\sqrt{9.2^2.3}=2\sqrt{27}>2\sqrt{16}\)
c)
\(\frac{1}{4}\sqrt{82}=\sqrt{\frac{82}{16}}=\sqrt{\frac{41}{8}}=\sqrt{5+\frac{1}{8}}\)
\(6\sqrt{\frac{1}{7}}=\sqrt{\frac{36}{7}}=\sqrt{5+\frac{1}{7}}\)
\(\sqrt{5+\frac{1}{8}}< \sqrt{5+\frac{1}{7}}\Rightarrow \frac{1}{4}\sqrt{82}< 6\sqrt{\frac{1}{7}}\)
d)
\(\frac{1}{2}\sqrt{\frac{17}{2}}=\sqrt{\frac{17}{8}}=\sqrt{2+\frac{1}{8}}\)
\(\frac{1}{3}\sqrt{19}=\sqrt{\frac{19}{9}}=\sqrt{2+\frac{1}{9}}\)
\(\sqrt{2+\frac{1}{8}}>\sqrt{2+\frac{1}{9}}\Rightarrow \frac{1}{2}\sqrt{\frac{17}{2}}> \frac{1}{3}\sqrt{19}\)
e)
\(3\sqrt{3}-2\sqrt{2}=\sqrt{27}-\sqrt{8}\)
Mà \(\sqrt{27}>\sqrt{25}; \sqrt{8}< \sqrt{9}\Rightarrow \sqrt{27}-\sqrt{8}> \sqrt{25}-\sqrt{9}=5-3=2\)
Vậy \(3\sqrt{3}-2\sqrt{2}>2\)
f)
\(\sqrt{7}+\sqrt{5}< \sqrt{9}+\sqrt{9}=6\)
\(\sqrt{49}=7\)
\(\Rightarrow \sqrt{7}+\sqrt{5}< 6< 7=\sqrt{49}\)
g)
\(\sqrt{2}< \sqrt{3}; \sqrt{11}< \sqrt{25}=5\)
\(\Rightarrow \sqrt{2}+\sqrt{11}< \sqrt{3}+5\)
h) Lặp lại câu d
i)
\(\sqrt{21}>\sqrt{20}\); \(\sqrt{5}< \sqrt{6}\)
\(\Rightarrow \sqrt{21}-\sqrt{5}> \sqrt{20}-\sqrt{6}\)
j) Lặp lại câu c
k)
\(\sqrt{6}>\sqrt{1}=1\)
\(\sqrt{20}>\sqrt{5}\)
\(\Rightarrow \sqrt{6}+\sqrt{20}>1+\sqrt{5}\)
\(\Rightarrow \sqrt{\sqrt{6}+\sqrt{20}}>\sqrt{1+\sqrt{5}}\)
l)
\(\sqrt{7}-\sqrt{2}=\sqrt{(\sqrt{7}-\sqrt{2})^2}=\sqrt{9-2\sqrt{14}}\)
Mà \(\sqrt{14}< \sqrt{16}=4\Rightarrow 2\sqrt{14}< 8\)
\(\Rightarrow 9-2\sqrt{14}>9-8=1\)
\(\Rightarrow \sqrt{7}-\sqrt{2}=\sqrt{9-2\sqrt{14}}>\sqrt{1}=1\)
m)
\(\sqrt{30}-\sqrt{29}=\frac{30-29}{\sqrt{30}+\sqrt{29}}=\frac{1}{\sqrt{30}+\sqrt{29}}\)
\(\sqrt{29}-\sqrt{28}=\frac{29-28}{\sqrt{29}+\sqrt{28}}=\frac{1}{\sqrt{29}+\sqrt{28}}\)
Mà \(\sqrt{30}+\sqrt{29}> \sqrt{29}+\sqrt{28}\Rightarrow \frac{1}{\sqrt{30}+\sqrt{29}}< \frac{1}{\sqrt{29}+\sqrt{28}}\)
\(\Rightarrow \sqrt{30}-\sqrt{29}< \sqrt{29}-\sqrt{28}\)
n)
\(\sqrt{8}+\sqrt{5}=\sqrt{(\sqrt{8}+\sqrt{5})^2}=\sqrt{13+2\sqrt{40}}\)
\(\sqrt{7}+\sqrt{6}=\sqrt{(\sqrt{7}+\sqrt{6})^2}=\sqrt{13+2\sqrt{42}}\)
Mà \(\sqrt{13+2\sqrt{40}}< \sqrt{13+2\sqrt{42}}\)
\(\Rightarrow \sqrt{8}+\sqrt{5}< \sqrt{7}+\sqrt{6}\)
o)
\(\sqrt{27}+\sqrt{6}+1>\sqrt{25}+\sqrt{4}=5+2=7\)
\(\sqrt{48}< \sqrt{49}=7\)
\(\Rightarrow \sqrt{27}+\sqrt{6}+1>\sqrt{48}\)
p)
\(5\sqrt{2}+\sqrt{75}=5\sqrt{2}+\sqrt{5^2.3}=5\sqrt{2}+5\sqrt{3}\)
\(5\sqrt{3}+\sqrt{50}=5\sqrt{3}+\sqrt{5^2.2}=5\sqrt{3}+5\sqrt{2}\)
\(\Rightarrow 5\sqrt{2}+\sqrt{75}=5\sqrt{3}+\sqrt{50}\)
q)
\(\sqrt{5}-\sqrt{3}=\frac{5-3}{\sqrt{5}+\sqrt{3}}=\frac{2}{\sqrt{(\sqrt{5}+\sqrt{3})^2}}=\frac{2}{\sqrt{8+2\sqrt{15}}}\)
\(8+2\sqrt{15}< 8+2\sqrt{16}=16\)
\(\Rightarrow \sqrt{8+2\sqrt{15}}< 4\)
\(\Rightarrow \sqrt{5}-\sqrt{3}=\frac{2}{\sqrt{8+2\sqrt{15}}}>\frac{2}{4}=\frac{1}{2}\)
