{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-power-root"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-power-root-01","title":{"zh-Hant":"自然數冪與主平方根：由遞迴算冪，並證明 √(a²)=|a|","en":"Natural-number powers and principal square roots: computing powers by recursion and proving √(a²)=|a|"},"summary":{"zh-Hant":"只用遞迴定義 a^0=1、a^(n+1)=a^n·a 算出 (−2)^n（n=0,1,2,3,4）與 0^0、0^3；證明非負平方根的唯一性，並證明對每個實數 a，√(a²)=|a|；再求 √49、√((−5)²)、方程 y²=9 的實數解集，並說明 √(−4) 在實數中沒有定義。","en":"Using only the recursive definition a^0=1, a^(n+1)=a^n·a, compute (−2)^n for n=0,1,2,3,4 and 0^0, 0^3; prove that a nonnegative square root is unique and that √(a²)=|a| for every real a; then find √49, √((−5)²) and the real solution set of y²=9, and explain why √(−4) is not defined in the reals."},"mko_ids":["mko-power-root"]}],"title":"概念推導導讀"}