<prompt> Mlinganyofrac1x frac1x 2 - frac1x 4 - frac1x 6 - frac1101 x 8 - frac1x 10 frac1x 12 frac1x 14 0ina mizizi minne ya umbo -a pm sqrtb pm c sqrtd, ambapo a, b, c, d ni nambari kamili chanya, na d haiwezi kugawanywa kwa mraba wa prime. . Pata a b c d.  <response>: Tunaweza kuoanisha masharti kama ifuatavyo:kushoto frac1x frac1x 14 kulia kushoto frac1x 2 frac1x 12 kulia - kushoto frac1x 4 frac1x 10 kulia - kushoto frac1x 6 frac 1x 8 kulia 0.Kishafrac2x 14x2 14x frac2x 14x2 14x 24 - frac2x 14x2 14x 40 - frac2x 14x2 14x 48 0.Tukigawanya kwa 2, tunapatafracx 7x2 14x fracx 7x2 14x 24 - fracx 7x2 14x 40 - fracx 7x2 14x 48 0.Acha y x 7. Kishafracyy2 - 49 fracyy2 - 25 - fracyy2 - 9 - fracyy2 - 1 0.Tunaona kuwa y 0 ni suluhisho. Vinginevyo, y neq 0, ili tuweze kugawanya pande zote mbili kwa y:frac1y2 - 49 frac1y2 - 25 - frac1y2 - 9 - frac1y2 - 1 0.Sasa, acha z y2, hivyofrac1z - 49 frac1z - 25 - frac1z - 9 - frac1z - 1 0. Kishafrac1z - 49 - frac1z - 9 frac1z - 1 - frac1z - 25.Kuchanganya sehemu kwa kila upande, tunapatafrac40z - 49z - 9 -frac24z - 1z - 25.Kwa hivyo, 40z - 1 z - 25 -24z - 49z - 9. Hii hurahisisha hadi z2 - 38z 181 0. Kwa fomula ya quadratic,z 19 pm 6 sqrt5.Kisha y pm sqrt19 pm 6 sqrt5, nax -7 pm sqrt19 pm 6 sqrt5.Hivyo, a b c d 7 19 6 5 37. 