text: # 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.
  
  ### Access hundreds of thousands of answers with a free trial.
  
  Ask a Question
  Related Questions
  Popular Questions
explanation: The document is an educational text that explains how to determine if the k-th root of a natural number n is an integer or irrational. It provides a clear explanation and examples to illustrate the concept. The text does not contain a specific question or a question-and-answer format but rather presents information in an explanatory manner.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
  
  ### Access hundreds of thousands of answers with a free trial.
  
  Ask a Question
  Related Questions
  Popular Questions
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an educational text that explains how to determine if the k-th root of a natural number n is an integer or irrational. It provides a clear explanation and examples to illustrate the concept. The text does not contain a specific question or a question-and-answer format but rather presents information in an explanatory manner.
  
  Label: no-question