<prompt> Acha x, y, z ziwe nambari halisi chanya. Pata thamani ya chini yafrac1 5z4z 3x5x 6yy 18xyz.  <response>: Tunaandikaanzaalignfrac1 5z4z 3x5x 6yy 18xyz frac45 cdot frac1 5z5z frac154 x5x 6yy 18xyz frac45 cdot frac43 cdot frac1 5z5z frac154 xfrac15 4 z frac92 yy 18xyz frac45 cdot frac43 cdot frac29 cdot frac1 5z5z frac154 xfrac154 x frac92 yfrac92 y 81xyz frac32135 cdot frac1 5z5z frac154 xfrac154 x frac99 2 yfrac92 y 81xyz.endalignHebu a 5z, b frac154 x, na c frac92 y, hivyo z frac1 5 a, x frac415 b, na y frac29 c. Kishaanzaalignfrac32135 cdot frac1 5z5z frac154 xfrac154 x frac92 yfrac92 y 81xyz frac32135 cdot frac1 aa bb c c 81frac415 b cdot frac29 c cdot frac15 a 20 cdot frac1 aa bb cc 81abc 20 cdoti 1 a kushoto 1 fracba kulia kushoto 1 fraccb kulia kushoto 1 frac 81c kulia.mwishoalignNa AM-GM,anzaalign1 a 1 fraca3 fraca3 fraca3 ge 4 sqrt4left fraca 3 kulia3, 1 fracba 1 fracb3a fracb3a fracb3a ge 4 sqrt4 kushoto fracb3a kulia3, 1 fraccb 1 fracc3b fracc3b fracc3b ge 4 sqrt4 kushoto fracc3b kulia3, 1 frac81c 1 frac27c frac27c frac27c ge 4 sqrt4 kushoto frac27c kulia3,mwishoalignhivyoanzaalign20 cdot 1 a kushoto 1 fracba kulia kushoto 1 fraccb kulia kushoto 1 frac81 c kulia ge 20 cdot 256 sqrt4left fraca3 kulia3 cdot left fracb3a kulia 3 cdoti kushoto fracc3b kulia3 cdoti kushoto frac27c kulia3 5120.endalignUsawa hutokea wakati1 fraca3 fracb3a fracc3b frac27c,au a 3, b 9, na c 27, ambayo ina maana x frac125, y 6, na z frac35. Kwa hiyo, thamani ya chini ni 5120. 