{ localUrl: '../page/566.html', arbitalUrl: 'https://arbital.com/p/566', rawJsonUrl: '../raw/566.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: '566', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'The n-th root of m is either an integer or irrational', clickbait: 'In other words, no power of a rational number that is not an integer is ever an integer.', textLength: '882', alias: '566', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JoeZeng', editCreatedAt: '2016-07-06 23:33:03', pageCreatorId: 'JoeZeng', pageCreatedAt: '2016-07-06 23:33:03', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '1', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '35', text: 'There is an intuitive way to see that for any natural numbers $m$ and $n$, $\\sqrt[n]m$ will always either be an integer or an irrational number.\n\nSuppose that there was some $\\sqrt[n]m$ that was a rational number $\\frac{a}{b}$ that was not an integer. Suppose further that $\\frac ab$ is written as a [-reduced] fraction, such that the [-greatest_common_divisor] of $a$ and $b$ is $1$. Then, since $\\frac{a}{b}$ is not an integer, $b > 1$.\n\nSince $\\frac ab = \\sqrt[n]m$, we have conversely that $(\\frac ab)^n = m$, which is a natural number by our hypothesis. But let's take a closer look at $(\\frac ab)^n$. It evaluates to $\\frac{a^n}{b^n}$, which is still a reduced fraction. [todo: Proof of gcd(a^n, b^n) = 1.]\n\nBut since $b > 1$ before, we have that $b^n > 1$ as well, meaning that $(\\frac ab)^n$ cannot be an integer, contradicting the fact that it equals $m$, a natural number.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '2', maintainerCount: '2', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '3', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JoeZeng', 'SharmaKunapalli' ], childIds: [], parentIds: [ 'math' ], commentIds: [], questionIds: [], tagIds: [ 'proof_meta_tag', 'needs_parent_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '23108', pageId: '566', userId: 'SharmaKunapalli', edit: '3', type: 'newEditProposal', createdAt: '2018-11-03 18:37:05', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16095', pageId: '566', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-07 23:33:50', auxPageId: 'needs_parent_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16094', pageId: '566', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-07 23:33:41', auxPageId: 'proof_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16092', pageId: '566', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-07 23:33:30', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15862', pageId: '566', userId: 'JoeZeng', edit: '1', type: 'newEdit', createdAt: '2016-07-06 23:33:03', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }