Question

Compare.

`root(3)(3)` and `root(4)(5)`

Answer 1

Given: Compare `root(3)(3)` and `root(4)(5)`

Stepwise calculation:

1. Express each root as an exponent with fractional power: 

`root(3)(3) = 3^(1/3)`,

`root(4)(5) = 5^(1/4)`.

2. Find a common denominator for the exponents to compare easily.

The LCM of 3 and 4 is 12. 

`3^(1/3) = 3^(4/12)`

= `(3^4)^(1/12)`

= `81^(1/12)`,

`5^(1/4) = 5^(3/12)`

= `(5^3)^(1/12)`

= `125^(1/12)`.

3. Since `(81 < 125), 81^(1/12) < 125^(1/12)`

So, `root(4)(5) > root(3)(3)`.