问题1：如果x + 1 / x = -2，则x 1000 + 1 / x 1000的值是
解决方案。 x = -1满足上述公式。
因此，(-1) 1000 +1 /(-1) 1000 = 1 +1 = 2
问题2：如果a = 1 – 1 / b和b = 1 – 1 / c，则c – 1 / a的值为
解决方案。设a = 2和b = -1
2 = 1 –(-1)
2 = 2正确
因此，找到c的值
b = 1 – 1 / c
-1 = 1 – 1 / c
c = 1/2
然后，c – 1 / a = 1/2 – 1/2 = 0
问题3：如果a + b + c = 3，则值1 /(1 – a)(1 – b)+ 1 /(1 – b)(1 – c)+ 1 /(1 – c)( 1 – a)
解决方案1 /(1 – a)(1 – b)+ 1 /(1 – b)(1 – c)+ 1 /(1 – c)(1 – a)
=> [(1 – c)+(1 – a)+(1 – b)] /(1 – a)(1 – b)(1 – c)
=> [3 –(a + b + c)] /(1 – a)(1 – b)(1 – c)
=> 3 – 3 /(1 – a)(1 – b)(1 – c)
=> 0
问题4：如果a + 1 / a =＆Sqrt; 3，则a 18 + a 12 + a 6 + 1的值是
解决方案。 a 3 + 1 / a 3 =(a + 1 / a) 3 – 3(a + 1 / a)
=> 3＆Sqrt; 3 – 3＆Sqrt; 3
=> 0
a 3 + 1 / a 3 = 0
6 + 1 = 0
然后，
18 + 12 + 6 + 1
a 12 (a 6 +1)+(a 6 +1)
12 x 0 + 0 = 0
问题5：如果a =＆Sqrt; 3 +1 /＆Sqrt; 3 -1和b =＆Sqrt; 3 -1 /＆Sqrt; 3 +1，则找到
(a 2 + ab + b 2 )/(a 2 – ab + b 2 )是
解决方案。 a = 1 / b
因此ab = 1
a + b =(＆Sqrt; 3 +1)/(＆Sqrt; 3 -1)+(＆Sqrt; 3 -1)/(＆Sqrt; 3 +1)
=>(3 +1 + 2＆Sqrt; 3 + 3 +1 – 2＆Sqrt; 3)/(3 – 1)
=> 8/2
=> 4
a + b = 4
a 2 + b 2 = 4 2 – 2 *(ab)
a 2 + b 2 = 14
现在，(a 2 + ab + b 2 )/(a 2 – ab + b 2 )
=>(14 +1)/(14 -1)
=> 15/13
问题6：如果x = 8，则求出x 5 – 9x 4 + 9x 3 – 9x 2 + 9x 1 – 1的值
解决方案。我们可以写成
8 5 – 8 * x 4 – 1 * x 4 + 8 * x 3 + 1 * x 3 – 8 * x 2 – 1 * x 2 + 8 * x 1 + 1 * x 1 – 1
现在把x = 8
8 5 – 8 * 8 4 – 1 * 8 4 + 8 * 8 3 + 1 * 8 3 – 8 * 8 2 – 1 * 8 2 + 8 * 8 1 + 1 * 8 1 – 1
= 8 – 1
= 7
问题7：如果两个实数的平方和为74，而它们的和为12，则这两个数的立方的总和为
解决方案让两个数字分别为a和b
给定a 2 + b 2 = 74
a + b = 12
(a + b) 2 = a 2 + b 2 + 2ab
12 2 = 74 + 2ab
144 = 74 + 2ab
ab = 35
我们得到a = 7和b = 5
然后，a 3 + b 3 = 7 3 + 5 3
= 343 + 125
= 468
问题8：如果ab(a + b)= m，则a 3 + b 3 + 3 m的值是
解决方案。 ab(a + b)= m
a + b = m / ab
双方的立方
a 3 + b 3 + 3ab(a + b)= m 3 / a 3 b 3
a 3 + b 3 + 3 m = m 3 / a 3 b 3
问题9：如果m =＆Sqrt; 7 +＆Sqrt; 7 +＆Sqrt; 7…..和n =＆Sqrt; 7-＆Sqrt; 7-＆Sqrt; 7……。
那么在m和n保持之间的以下关系中
解决方案。 m =＆Sqrt;(7 + m)
m 2 = 7 + m
m 2 – m = 7……。(1)
并且n =＆Sqrt;(7 – n)
n 2 + n = 7……。(2)
来自(1)和(2)
m 2 – m = n 2 + n
m 2 – n 2 –(m + n)= 0
(m + n)(m – n)–(m + n)= 0
m – n – 1 = 0
问题10：如果x 2 + y 2 + z 2 = 2(x + y -1)，那么x 3 + y 3 + z 3的值是多少?
解决方案。 x 2 + y 2 + z 2 = 2x + 2y -2
(x 2 +1 -2x)+(y 2 +1 -2y)+(z 2 )= 0
(x – 1) 2 +(y – 1) 2 +(z) 2 = 0
=>(x – 1) 2 = 0
=> x = 1
(y – 1) 2 = 0
=> y = 1
(z) 2 = 0
=> z = 0
将价值放在等式中
x 3 + y 3 + z 3
1 3 +1 3 + 0 3
=> 2
问题11：如果(x 12 +1)/ x 6 = 6，那么(x 36 +1)/ x 18的值是多少?
解决方案。给定
(x 12 +1)/ x 6 = 6
x 6 +1 / x 6 = 6
双方的立方
(x 6 +1 / x 6 ) 3 = 6 3
x 18 + 1 / x 18 + 3(x 6 +1 / x 6 )= 216
x 18 + 1 / x 18 + 3 * 6 = 216
x 18 + 1 / x 18 = 198
(x 36 +1)/ x 18 = 198