Question

I want to know how to compare in root. Express large and small. `A = 3^(1/3), B = 4^(1/4), C = 6^(1/6)` ?

Express large and small.
`A = 3^(1/3), B = 4^(1/4), C = 6^(1/6)`

Answer 1

`A> B > C`

Explanation:

Just take the logarithm (to the base `e` or `10`)

In the given case let us take logarithm to the base`10`

`logA=1/3log3=1/3xx0.4771~=0.1590`

`logB=1/4log4=1/4xx0.6021~=0.1505`

`logC=1/6log6=1/6xx0.7781~=0.1297`

As `logA>logB>logC`, we have

`A> B > C`

This is as `logx` function is continuously increasing function, as seen from the following graph; as also derivative of `logx` is `1/x`, which is negative only for `x<0`, which is not in the domain.

graph{logx [-0.29, 9.71, -2.22, 2.78]}