一個數學問題─指數

Question

1. x^1/2+x^-1/2=3,則x^3/2+x^-3/2=?

2. x^1/2+x^-1/2=3,則x^3/2+x^-3/2+3/x^2+x^-2+5=?

3. a^1/2+a^-1/2=4,則(a^1/2-a^-1/2)^2+3=?

一位用功的高三生

Answer 1

公式:a3+b3=(a+b) 3-3ab(a+b) x3+(1/x)3=[x+(1/x)] 3-3[x*(1/x)]*[x+(1/x)] =[x+(1/x)] 3-3 [x+(1/x)] 1. x^1/2+x^-1/2=3,則x^3/2+x^-3/2=? [解] x3/2+x-3/2=[x1/2+(1/x) 1/2] 3-3 [x1/2+(1/x) 1/2]=3 3-3*3=18 2. x^1/2+x^-1/2=3,則x^3/2+x^-3/2+3/x^2+x^-2+5=? [解]題目有點不明!x+x-1=( x1/2+x-1/2) 2-2 x1/2*x-1/2=32-2=7x2+x2=(x+x-1) 2-2 x*x-1=72-2=47x3/2+x-3/2=[x1/2+(1/x) 1/2] 3-3 [x1/2+(1/x) 1/2]=3 3-3*3=18x3/2+x-3/2+3/[ (x2+x2)+5] = 18+3/(47+5)=18+(1/14)=18又1/14 3. a1/2+a-1/2=4,則(a1/2-a-1/2) 2+3=?[解] 公式(a-b)2=a2-2ab+b2=(a2+2ab+b2)-2ab-2ab =(a+b) 2-4ab(a1/2-a-1/2) 2+3=(a1/2+a-1/2) 2-4a1/2*a-1/2+3=42-4+3=15

Answer 2

首先，你要知道的公式有：
x^3 + y^3 = (x+y)(x^2 - xy + y^2)
(x - y)^2 = (x + y)^2 - 4xy
(x + y)^2 = x^2 + 2xy +y^2
然後我們現在把 x^1/2 當做一個變數，則x^3/2 = (x^1/2)^3

解：
1. x^1/2+x^-1/2=3，則 9 = (x^1/2+x^-1/2)^2 = x + 2 + x^-1 => x + x^-1 = 7
則 x^3/2+x^-3/2 = (x^1/2+x^-1/2) [x - x^(1/2)*x^(-1/2) + x^-1]
= 3 * ( 7 - 1) = 18

因為第2題不知道你題目是不是我知道的。
2.
x^1/2+x^-1/2=3，則 x + x^-1 = 7 ，x^3/2+x^-3/2 = 18 (第一題算過)，
x^2 + x^-2 = 47 (和x + x^-1 = 7同樣的算法，你可以自己試試看)
所求如果是： x^3/2+x^-3/2+3/(x^2+x^-2+5)
= 18 + 3/ (47 + 5) = 18 + 3/52 = 939/52 (你也可以用帶分數表示)

3. a^1/2+a^-1/2=4，
則(a^1/2 - a^-1/2)^2 = (a^1/2 + a^-1/2)^2 - 4*a^(1/2) *a^(-1/2) = 16 - 4 = 12
所求：(a^1/2-a^-1/2)^2 + 3= 12 + 3 = 15