Question

Cmr: A = \(\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\) là số nguyên

Answer 1

=A=\(\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)

có (a+b)³=a³+3a²b+3ab²+b³

=a³+b³+3ab(a+b)

Ad ta có

A³=2+3(\(\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)) .

(\(\sqrt[3]{\left(1+\dfrac{\sqrt{84}}{9}\right)\left(1-\dfrac{\sqrt{84}}{9}\right)}\))

A³=2+3A\(\sqrt[3]{1-\dfrac{84}{81}}\)

A³=2-A

=>A³+A-2=0

=>A³-A+2A-2=0

=>A(A²-1)+2(A-1)=0

=>A(A-1)(A+1)+2(A-1)=0

=>(A-1)(A²+A+2)=0

=>(A-1)(A²+2.\(\dfrac{1}{2}\)A+\(\dfrac{1}{4}\)+\(\dfrac{7}{4}\))=0

^=>(A-1)((A+\(\dfrac{1}{2}\))²+\(\dfrac{7}{4}\))=0

=> A=1

hoặc (A+\(\dfrac{1}{2}\))²+\(\dfrac{7}{4}\)=0(loại)

vậy A nguyên