<prompt> Tuseme kwamba x, y, na z ni nambari changamano kama vile anzailiyolingana xy -80 - 320i, yz 60, zx -96 24i , mwishozilizopangiliwaambapo i2 -1. Kokotoa xyz.  <response>: Kuzidisha milinganyo iliyotolewa pamoja, tuna anzailiyopangwa xyz2 -80-320i cdot 60 cdot -9624i -8014i cdot 60 cdot -244-i 80 cdot 60 cdot 24 8 15i. endalignedIli kutatua kwa xyz, tunapata nambari changamano abi ambayo mraba wake ni 815i ambapo a na b ni halisi; yaani tunataka abi2 a2-b2 2abi 8 15i.Tukilinganisha sehemu halisi na za kufikirika, tunapata milinganyo a2-b. 28 na 2ab15. Kisha b frac152a; kubadilisha katika mlinganyo mwingine inatoa a2 - frac2254a2 8, au 4a4 - 32a2 - 225 0. Sababu hizi kama 2a2-252a29 0hivyo 2a2-250 kwa kuwa a ni halisi, na a pm frac5sqrt2. Kisha b frac152a pm frac3sqrt2. Kwa hivyo, kiholela kuchagua a na b chanya, tuna xyz2 80 cdot 60 cdot 24 leftfrac5sqrt2 frac3sqrt2i kulia2,na hivyo anzaaligned xyz pm sqrt80 cdot 60 cdot 24leftfrac5 sqrt2 frac3sqrt2i kulia pm240sqrt2 kushotofrac5sqrt2 frac3sqrt2 i kulia pm24053i. endalignedKisha anzaaligned x fracxyzyz pm frac 24053i60 pm 20 12i , z fracxyzxy pm frac 24053i-8014i pm frac-3 53i 1-4i17 pm frac -317-17i17 pm -33i, y fracxyzxz pmfrac 24053i-244-i pm frac- 10 53i4i17 pm frac -101717i17 pm-10-10i. mwishozilizopangiliwaKwa hivyo, xyz pm7 5i, hivyo xyz sqrt7252 sqrt74. 