Đặt giá trị của biểu thức \(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\) = x   (1)

Lập phương cả 2 vế , ta có: 

x3 = (\(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\))3

x3 = \(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)3 + 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)2(\(\sqrt[3]{8-3\sqrt{21}}\)) + 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)(\(\sqrt[3]{8-3\sqrt{21}}\))2+(\(\sqrt[3]{8-3\sqrt{21}}\))3

x3 = 8 + \(3\sqrt{21}\)+ 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)+ 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)+ \(8-3\sqrt{21}\)

x3 = 16 + 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)(\(\sqrt[3]{\left(8+3\sqrt{21}\right)\left(8-3\sqrt{21}\right)}\)) + 3\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)(\(\sqrt[3]{\left(8+3\sqrt{21}\right)\left(8-3\sqrt{21}\right)}\))

x3 = 16 + 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)(\(\sqrt[3]{-125}\)) + 3\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)(\(\sqrt[3]{-125}\))

x3 = 16 + 3\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\)( -5 ) + 3\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)( -5 )

x3 = 16 - 15\(\left(\sqrt[3]{8+3\sqrt{21}}\right)\) - 15\(\left(\sqrt[3]{8-3\sqrt{21}}\right)\)

x3 = 16 - 15(\(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\))

Theo (1) ,ta có: x3 = 16 - 15x ⇔ x3 + 15x = 16

Ta thấy chỉ có x = 1 là phù hợp (2)

Thay (2) vào (1) ⇒ \(\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}\) = 1

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