Question

So sánh các cặp số sau :

1. 4\(\sqrt{7}\) và 3\(\sqrt{3}\)

2. \(\dfrac{1}{4}\sqrt{82}\) và \(6\sqrt{\dfrac{1}{7}}\)

3. x = 3+\(\sqrt{2}\) : y=\(\sqrt{11+6\sqrt{2}}\)

Tính x² ; y² => So sánh x và y

Answer 1

3.

\(•x=3+\sqrt{2}\\ x^2=\left(3+\sqrt{2}\right)^2\\ x^2=9+2.3.\sqrt{2}+2\\ x^2=11+6\sqrt{2}\\• y=\sqrt{11+6\sqrt{2}}\\ y^2=\left(\sqrt{11+6\sqrt{2}}\right)^2\\ y^2=11+6\sqrt{2}\)

\(\Rightarrow x^2=y^2=11+6\sqrt{2}\)

Answer 2

1. ta có : \(4\sqrt{7}=\sqrt{112}\)

\(3\sqrt{3}=\sqrt{27}\)

ta thấy : \(\sqrt{112}>\sqrt{27}\) hay \(4\sqrt{7}>3\sqrt{3}\)

2. \(\dfrac{1}{4}\sqrt{82}=\sqrt{\dfrac{41}{8}}\)

\(6\sqrt{\dfrac{1}{7}}=\sqrt{\dfrac{36}{7}}\)

ta thấy :\(\sqrt{\dfrac{41}{8}}< \sqrt{\dfrac{36}{7}}\) hay \(\dfrac{1}{4}\sqrt{82}< 6\sqrt{\dfrac{1}{7}}\)

3. \(x^2=\left(3+\sqrt{2}\right)^2\)

\(y^2=11+6\sqrt{2}\)=\(\left(3+\sqrt{2}\right)^2\)

ta thấy : \(x^2=y^2\Rightarrow x=y\)