Question

Compare.

`root(4)(7)` and `root(3)(6)`

Answer 1

Given: Compare `root(4)(7)` and `root(3)(6)`

Stepwise calculation:

1. Rewrite each root in exponential form:

`root(4)(7) = 7^(1/4)`,

`root(3)(6) = 6^(1/3)`

2. To compare these, raise both to the power of 12 the least common multiple of 3 and 4 to avoid dealing with fractional powers directly:

`(7^(1/4))^12 = 7^3 = 343`,

`(6^(1/3))^12 = 6^4 = 1296`

3. Since (343 < 1296), it follows that:

`(7^(1/4)) < (6^(1/3))` or equivalently, `root(4)(7) < root(3)(6)`

Thus, the cube root of 6 is greater than the fourth root of 7.