# How can you determine whether the k-th root of n is an integer or irrational with k and n being natural?

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### 1 Answer

The k-th root of a natural number n is an integer if n can be expressed as an integer m raised to the power k. If n = m^k, the k-th power of n is an integer: `n^(1/k) = (m^k)^(1/k) = m`

For example ``the third root of 8 is an integer as 8 can be written as 2^3.

The k-th power of a natural number is irrational if it is not possible to write n in the form m^k. For example, the `sqrt 2` is irrational as 2 cannot be written as the square of an integer.

An irrational number is one that cannot be expressed in the form p/q where p and q are natural numbers that are mutually prime.