求解 831/81!+82!

$$\frac { 831 } { 81 ! + 82 ! }$$

求值

$$\frac{277}{160387153240677180942672647073661018583075217104241651525302266375486794971531013556421105447576304353280000000000000000000}\approx 1.727070993 \cdot 10^{-120}$$

求解步骤

81 的阶乘是 5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000。

$$\frac{831}{5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000+82!}$$

82 的阶乘是 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000。

$$\frac{831}{5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000+475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000}$$

5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000 与 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000 相加，得到 481161459722031542828017941220983055749225651312724954575906799126460384914593040669263316342728913059840000000000000000000。

$$\frac{831}{481161459722031542828017941220983055749225651312724954575906799126460384914593040669263316342728913059840000000000000000000}$$

通过求根和消去 3，将分数 \frac{831}{481161459722031542828017941220983055749225651312724954575906799126460384914593040669263316342728913059840000000000000000000} 降低为最简分数。

$$\frac{277}{160387153240677180942672647073661018583075217104241651525302266375486794971531013556421105447576304353280000000000000000000}$$

因式分解

$$\frac{277}{2^{78}\times 3^{39}\times 5^{19}\times 7^{12}\times 11^{7}\times 13^{6}\times 17^{4}\times 19^{4}\times 23^{3}\times 29^{2}\times 31^{2}\times 37^{2}\times 41\times 43\times 47\times 53\times 59\times 61\times 67\times 71\times 73\times 79\times 83}$$