Question

Chứng minh rằng: 

\(a)\ (\dfrac{1}{3})^{2\sqrt{5}}<(\dfrac{1}{3})^{3\sqrt{2}}\)

\(b)\ 7^{\sqrt[6]{3}}<7^{\sqrt[3]{6}}\)

 

Answer 1

a) ta có 2√5= = √20 ; 3√2 = = √ 18 => 2√5 > 3√2

=> <

b) 6√3 = = √108 ; 3√6 = = √54 => 6√3 > 3√6 => >

Answer 2

a) \(2\sqrt{5}=\sqrt{2^2.5}=\sqrt{20}\)

\(3\sqrt{2}=\sqrt{3^2.2}=\sqrt{18}\)

=> \(2\sqrt{5}>3\sqrt{2}\)

=> \(\left(\dfrac{1}{3}\right)^{2\sqrt{5}}< \left(\dfrac{1}{3}\right)^{3\sqrt{2}}\)

(vì cơ số \(\dfrac{1}{3}< 1\))

b) Vì \(3< 6^2\)

=> \(3^{\dfrac{1}{6}}< \left(6^2\right)^{\dfrac{1}{6}}\)

=> \(\sqrt[6]{3}< 6^{\dfrac{1}{3}}\)

=> \(\sqrt[6]{3}< \sqrt[3]{6}\)

=> \(7^{\sqrt[6]{3}}< 7^{\sqrt[3]{6}}\)