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Question Number 156419 by cortano last updated on 11/Oct/21

x is positive integer number

can you check if Q = \sqrt[3]{({(x + 2)}^{4} - x^{4}}

is a natural number

Commented by mr W last updated on 11/Oct/21

x = 2 :

Q = \sqrt[3]{4^{4} - 2^{4}} = \sqrt[3]{240} = 2 \sqrt[3]{30} \notin N

Answered by Rasheed.Sindhi last updated on 11/Oct/21

Q = \sqrt[3]{{(x + 2)}^{4} - x^{4}}

= \sqrt[3]{x^{4} + {8 x}^{3} + {24 x}^{2} + 32 x + 16 - x^{4}}

= \sqrt[3]{{8 x}^{3} + {24 x}^{2} + 32 x + 16}

= 2 \sqrt[3]{x^{3} + {3 x}^{2} + 4 x + 2}

= 2 \sqrt[3]{(x^{3} + {3 x}^{2} + 3 x + 1) + (x + 1)}

= 2 \sqrt[3]{{(x + 1)}^{3} + (x + 1)}

The only solution, I think, is :

x + 1 = 0

x = - 1 \in N

No natural x