Question

怎么比较2的1/2次方、3的1/3次方、4的1/4次方和5的1/5次方谁大谁小?

请问"√"是什么意思?

Answer 1

^表示乘方

2^(1/2)=2^(1/3*3/2)=[2^(3/2)]^1/3 = (2*√2)^1/3
2*√2=√8>√9=3
所以2^(1/2) = (2*√2)^1/3 > 3^(1/3)
即2^(1/2) > 3^(1/3)

4^(1/4)=2^(1/2)

5^(1/5)=5^(1/3*3/5)=[5^(3/5)]^1/3
5^(3/5)=(125)^(1/5)<(243)^(1/5)=3
所以5^(1/5) < 3^(1/3)

综上所述5^(1/5) < 3^(1/3) < 4^(1/4) = 2^(1/2)

还有一种方法就是构造函数然后求导，你可能没学过，如果你学过又想知道就再找我好了。