# 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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justaguide | Certified Educator

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.

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The text passage provides an explanation of how to determine if the k-th root of a natural number n is an integer or irrational. It includes examples and a definition of an irrational number.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# How can you determine whether the k-th root of n is an integer or irrational with k and n being natural?

Expert Answers
justaguide | Certified Educator

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.

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The text passage provides an explanation of how to determine if the k-th root of a natural number n is an integer or irrational. It includes examples and a definition of an irrational number.

TYPE: NO_QUESTION