Question

高一数学 分数指数幂求解

解方程
3^x+3^(1-x)=4
2^2x-6·2^(x+1)+2^5=0
求值题
若x＞0，y＞0，且x^1/2·(x^1/2+y^1/2)=3y^1/2·(x^1/2+5y^1/2)。求(2x+2(xy)^1/2+3y/(x-(xy)^1/2+y)的值

Answer 1

3^x+3^(1-x)=4
3^x+3/3^x=4
(3^x)^2-4*3^x=3=0
(3^x-1)(3^x-3)=0
3^x=1 3^x=3
x=0 x=1

2^2x-6·2^(x+1)+2^5=0
2^2x-12*2^x+32=0
(2^x-4)(2^x-8)=0
2^x-4=0 2^x-8=0
x=2 x=3

若x＞0，y＞0，且x^1/2·(x^1/2+y^1/2)=3y^1/2·(x^1/2+5y^1/2)
x+x^1/2y^1/2=3x^1/2y^1/2+15y
x-2x^1/2y^1/2-15y=0
(x^1/2-5y^1/2)(x^1/2+3y^1/2)=0
x^1/2=5y^1/2 x^1/2=-3y^1/2(不合题意，舍去）
x=25y
(2x+2(xy)^1/2+3y/(x-(xy)^1/2+y)
=(2*25y+2(25y*y)^1/2+3y/(25y-(25y*y)^1/2+y)
=(50y+10y+3y)/(25y-5y+y)
=63y/21y
=3

Answer 2

3^x+3^(1-x)=4 ==.> 3^2x -4*3^x+3 =0 ==>(3^x-1)(3^x-3)=0
x=0，x=1
2^2x-6·2^(x+1)+2^5=0 ==>2^2x-12*2^x+32=0 ==>(2^x-4)(2^x-8)=0
x=2，x=3
x^1/2·(x^1/2+y^1/2)=3y^1/2·(x^1/2+5y^1/2)
==>x=15y+2x^1/2*y^1/2
令：y=1，则：x=15+2x^1/2 ==>x=25
2x+2(xy)^1/2+3y/(x-(xy)^1/2+y) =421/7

Answer 3

解方程用换元法。
求值题将原式展开解得X=Y,代入可得结果。