Question

已知x+1÷x=3，求下列各代数式的值：

已知x+1÷x=3，求下列各代数式的值：（1）x^2+1÷x^2 ；（2）x^3+1÷x^3 ；
（3）x^4+1÷x^4 ；（4）x^5+1÷x^5
要详细的过程！

Answer 1

x+1÷x=3 两边平方 得 x^2+1÷x^2+2x（1÷x）=9 则：x^2+1÷x^2=7 以此类推

Answer 2

(1) x^2+1/x^2=(x+1/x)^2-2=3^2-2=7
(2) x^3+1/x^3=(x+1/x)*(x^2-1+1/x^2)=(x+1/x)*[(x+1/x)^2-3]=3*[3^2-3]=18
(3) x^4+1/x^4=[x^2+1/x^2]^2-2=[3^2-2]^2-2=47
(4)x^5+1/x^5=[x^2+1/x^2]*[ x^3+1/x^3]-(x+1/x)=7*18-3=123

Answer 3

(1)∵x+1÷x=3
∴x²+1÷x²+2=9
∴x^2+1÷x^2 =7
(2)x^3+1÷x^3 =(x+1/x)（x²-1+1÷x²)
=3×（7-1）=18
（3）x^4+1÷x^4 =（x^2+1÷x^2)²-2=7²-2=47
(4)∵(x+1÷x)(x^4+1÷x^4)=x^5+1÷x^5 +x^3+1÷x^3
∴x^5+1÷x^5 =3×47-18=123

Answer 4

最快的方法是解得1/x=2
x^2+1÷x^2 =1+1/x^2=1+4=5
x^3+1÷x^3=1+8=9
x^4+1÷x^4 =17
x^5+1÷x^5 =33

Answer 5

7 18 47 123