一道有关分数指数幂的数学题
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一道有关分数指数幂的数学题
已知x+x^(-1)=4,求下列各式的值
(1)x^(1/2)+x^(-1/2)
(2) x^(3/2)+x^(-3/2)
(3) x^(1/2)-x^(-1/2)
已知x+x^(-1)=4,求下列各式的值
(1)x^(1/2)+x^(-1/2)
(2) x^(3/2)+x^(-3/2)
(3) x^(1/2)-x^(-1/2)
![一道有关分数指数幂的数学题](/uploads/image/z/13590710-62-0.jpg?t=%E4%B8%80%E9%81%93%E6%9C%89%E5%85%B3%E5%88%86%E6%95%B0%E6%8C%87%E6%95%B0%E5%B9%82%E7%9A%84%E6%95%B0%E5%AD%A6%E9%A2%98)
已知x+x^(-1)=4,两边平方:x²+2+x^(-2)=16,得x²+x^(-2)=14
(1) [x^(1/2)+x^(-1/2)]²=x+2+x^(-1)=4+2=6,
故x^(1/2)+x^(-1/2)=√6 .
(2) [x^(3/2)+x^(-3/2)]²=x^3+x^(-3)+2=[x+x^(-1)][x²-1+x^(-2)]+2
=4×(14-1)+2=50,
故x^(3/2)+x^(-3/2)=√50 .
(3) [x^(1/2)-x^(-1/2)]²= x-2+x^(-1)=4-2=2,
故x^(1/2)-x^(-1/2)=±√2 .
(1) [x^(1/2)+x^(-1/2)]²=x+2+x^(-1)=4+2=6,
故x^(1/2)+x^(-1/2)=√6 .
(2) [x^(3/2)+x^(-3/2)]²=x^3+x^(-3)+2=[x+x^(-1)][x²-1+x^(-2)]+2
=4×(14-1)+2=50,
故x^(3/2)+x^(-3/2)=√50 .
(3) [x^(1/2)-x^(-1/2)]²= x-2+x^(-1)=4-2=2,
故x^(1/2)-x^(-1/2)=±√2 .