V algebri vkladenim radikalom nazivayut radikal sho mistitsya v inshomu radikali Napriklad 5 2 5 displaystyle sqrt 5 2 sqrt 5 abo skladnishij priklad 2 3 4 3 3 displaystyle sqrt 3 2 sqrt 3 sqrt 3 4 Znachennya vsih vkladenih radikaliv nazivayut virazni mi v radikalah Zmist 1 Sproshennya vkladenih radikaliv 2 Neskinchenno vkladeni radikali 2 1 Zagalni polozhennya 2 2 Trivialni vipadki 2 3 Netrivialni vipadki 2 4 Okremi vipadki 3 PosilannyaSproshennya vkladenih radikaliv RedaguvatiDeyaki vkladeni radikali mozhna sprostiti Napriklad 3 2 2 1 2 displaystyle sqrt 3 2 sqrt 2 1 sqrt 2 nbsp 2 3 1 3 1 2 3 4 3 9 3 displaystyle sqrt 3 sqrt 3 2 1 frac 1 sqrt 3 2 sqrt 3 4 sqrt 3 9 nbsp U zagalnomu vipadku sproshennya ye skladnoyu problemoyu yaksho vono vzagali mozhlive Koli R a 2 b 2 c displaystyle R sqrt a 2 b 2 c nbsp racionalne zrobiti sproshennya dozvolyaye taka formula a b c a R 2 a R 2 displaystyle sqrt a pm b sqrt c sqrt frac a R 2 pm sqrt frac a R 2 nbsp Napriklad a a 2 b 2 a b 2 a b 2 a b displaystyle sqrt a pm sqrt a 2 b 2 sqrt frac a b 2 pm sqrt frac a b 2 quad a geq b nbsp Zokrema dlya kompleksnih chisel c 1 displaystyle c 1 nbsp a b i z a 2 i sgn b z a 2 displaystyle sqrt a bi pm left sqrt frac left z right a 2 i operatorname sgn b sqrt frac left z right a 2 right nbsp de z a 2 b 2 displaystyle left z right sqrt a 2 b 2 nbsp Neskinchenno vkladeni radikali RedaguvatiZagalni polozhennya Redaguvati U deyakih vipadkah neskinchenno vkladeni radikali mozhut buti totozhnimi deyakomu racionalnomu chislu napriklad viraz x 2 2 2 2 displaystyle x sqrt 2 sqrt 2 sqrt 2 sqrt 2 cdots nbsp dorivnyuye 2 Dlya togo shob ce pobachiti pidnesedemo obidvi chastini virazu do kvadrata i vidnimemo 2 x 2 2 2 2 2 x displaystyle x 2 2 sqrt 2 sqrt 2 sqrt 2 cdots x nbsp x 2 x 2 0 displaystyle x 2 x 2 0 nbsp x 1 2 x 2 1 displaystyle x 1 2 x 2 1 nbsp Ochevidno sho 1 displaystyle 1 nbsp ne mozhe buti znachennyam vihidnogo radikala Trivialni vipadki Redaguvati Dlya kvadratnogo korenya a b a b a b a b b b 2 4 a 2 displaystyle sqrt a b sqrt a b sqrt a b sqrt a b sqrt cdots frac b sqrt b 2 4a 2 nbsp Dlya korenya stepenya n displaystyle n nbsp a b a b a b a b n n n n n x displaystyle sqrt n a b sqrt n a b sqrt n a b sqrt n a b sqrt n cdots x nbsp de x displaystyle x nbsp ye rozv yazkom rivnyannya x n b x a 0 displaystyle x n bx a 0 nbsp Netrivialni vipadki Redaguvati Formula Ramanudzhana x n a a x n a 2 x a x n n a 2 x n a x 2 n n a 2 x 2 n displaystyle x n a sqrt ax n a 2 x sqrt a x n n a 2 x n sqrt a x 2n n a 2 x 2n sqrt cdots nbsp Okremi vipadki Redaguvati Zolotij peretin ϕ 1 1 1 displaystyle phi sqrt 1 sqrt 1 sqrt 1 sqrt cdots nbsp Plastichne chislo r 1 1 1 3 3 3 3 displaystyle rho sqrt 3 1 sqrt 3 1 sqrt 3 1 sqrt 3 cdots nbsp Chislo pi 2 p 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 displaystyle frac 2 pi sqrt frac 1 2 sqrt frac 1 2 frac 1 2 sqrt frac 1 2 sqrt frac 1 2 frac 1 2 sqrt frac 1 2 frac 1 2 sqrt frac 1 2 cdots nbsp Posilannya RedaguvatiWeisstein Eric W Vkladeni radikali angl na sajti Wolfram MathWorld Weisstein Eric W Kvadratnij korin angl na sajti Wolfram MathWorld Decreasing the Nesting Depth of Expressions Involving Square Roots angl Simplifying Square Roots of Square Roots Arhivovano 15 sichnya 2022 u Wayback Machine angl Otrimano z https uk wikipedia org w index php title Vkladeni radikali amp oldid 36047598
