\(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\)=\(\dfrac{8+3\sqrt{21}+8-3\sqrt{21}}{\sqrt[3]{\left(8+3\sqrt{21}\right)^2}-\sqrt[3]{\left(8+3\sqrt{21}\right).\left(8-3\sqrt{21}\right)}+\sqrt[3]{\left(8-3\sqrt{21}\right)^2}}\)

                                          =   \(\dfrac{16}{\left(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\right)^2-3\sqrt[3]{\left(8^2-\left(3\sqrt{21}\right)^2\right)}}\)

Đặt a= \(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\)

khi đó ta có phương trình: a= \(\dfrac{16}{\left(a\right)^2+15}\)=> a³+15a-16=0 => a=1 

Vậy \(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\) = 1