根号19怎么化简

Question

根号19怎么化简

Answer 1

因为有二项式定理： (1 + x)^n = 1 + nx + n(n-1)/2 x² + n(n-1)(n-2)/6 x ³ + .........
当 n = 1/2时： √(1+x) = 1 + 1/2x + 1/2(-1/2)/2x² + .1/2(-1/2)(-3/2)/6x³ ..............
= 1 +1/2 x -1/8x² +1/16 x³ - 5/128 x⁴+ 7/256 x^5
√19 = √(16+3)
= 4√(1+3/16)
= 4 [1 +1/2 x -1/8x² +1/16 x³ - 5/128 x⁴+ 7/256 x^5 +......] 其中 x=3/16
下面分别给出不同精度的数值：
≈4[1+ 3/32] = 35 / 8 = 4.3750 取1阶
≈4[1+ 3/32 - 1/8 *3/16] = 2231 / 512 = 4.3574 取2阶
≈4[1+ 3/32 - 1/8 *3/16 +1/16*3/16] = 1129 / 259 = 4.3591 取3阶
≈4[1+ 3/32 - 1/8 *3/16 +1/16*3/16 - 5/128 x⁴] = 3413/783 = 4.3589 取4阶
≈4[1+ 3/32 - 1/8 *3/16 +1/16*3/16 - 5/128 x⁴+ 7/256 x^5] = 1591/365 = 4.3589 取5阶
真实值: √19 ≈ 4.35889894 可见取取3阶 精度已经足够了

Answer 2

没得化简，19是质数本回答被提问者采纳

Answer 3

这就是最简式啦。