Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

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

x is positive integer number can you check if Q=((((x+2)^4 −x^4 ))^(1/3) is a natural number

$$\mathrm{x}\:\mathrm{is}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{number}\: \\ $$$$\mathrm{can}\:\mathrm{you}\:\mathrm{check}\:\mathrm{if}\:\mathrm{Q}=\sqrt[{\mathrm{3}}]{\left(\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{4}} −\mathrm{x}^{\mathrm{4}} \right.} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{natural}\:\mathrm{number} \\ $$

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

x=2: Q=((4^4 −2^4 ))^(1/3) =((240))^(1/3) =2((30))^(1/3) ∉N

$${x}=\mathrm{2}: \\ $$$${Q}=\sqrt[{\mathrm{3}}]{\mathrm{4}^{\mathrm{4}} −\mathrm{2}^{\mathrm{4}} }=\sqrt[{\mathrm{3}}]{\mathrm{240}}=\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{30}}\notin{N} \\ $$

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

Q=(((x+2)^4 −x^4 ))^(1/3) =((x^4 +8x^3 +24x^2 +32x+16−x^4 ))^(1/3) =((8x^3 +24x^2 +32x+16))^(1/3) =2((x^3 +3x^2 +4x+2))^(1/3) =2(((x^3 +3x^2 +3x+1)+(x+1)))^(1/3) =2(((x+1)^3 +(x+1)))^(1/3) The only solution,I think,is: x+1=0 x=−1 ∈N No natural x

$$\:\mathrm{Q}=\sqrt[{\mathrm{3}}]{\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{4}} −\mathrm{x}^{\mathrm{4}} } \\ $$$$\:\:\:\:=\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{4}} +\mathrm{8x}^{\mathrm{3}} +\mathrm{24x}^{\mathrm{2}} +\mathrm{32x}+\mathrm{16}−\mathrm{x}^{\mathrm{4}} } \\ $$$$\:\:\:\:=\sqrt[{\mathrm{3}}]{\mathrm{8x}^{\mathrm{3}} +\mathrm{24x}^{\mathrm{2}} +\mathrm{32x}+\mathrm{16}} \\ $$$$\:\:\:\:=\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{3}} +\mathrm{3x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{2}} \\ $$$$\:\:\:\:=\mathrm{2}\sqrt[{\mathrm{3}}]{\left(\mathrm{x}^{\mathrm{3}} +\mathrm{3x}^{\mathrm{2}} +\mathrm{3x}+\mathrm{1}\right)+\left(\mathrm{x}+\mathrm{1}\right)} \\ $$$$\:\:\:\:=\mathrm{2}\sqrt[{\mathrm{3}}]{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} +\left(\mathrm{x}+\mathrm{1}\right)} \\ $$$$\mathrm{The}\:\mathrm{only}\:\mathrm{solution},\mathrm{I}\:\mathrm{think},\mathrm{is}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{x}+\mathrm{1}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}=−\mathrm{1}\:\in\mathbb{N} \\ $$$$\mathrm{No}\:\mathrm{natural}\:\mathrm{x} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com