text: How do you simplify sqrt (16) / (sqrt (4) + sqrt (2))?
  
  It is
  
  sqrt (16) / (sqrt (4) + sqrt (2))=4/[sqrt2(sqrt2+1)]= 2sqrt2/(sqrt2+1)=2*sqrt2(sqrt2-1)/[(sqrt2+1)*(sqrt2-1)]= 2sqrt2(sqrt2-1)
  
  Jun 26, 2016
  
  $4 - 2 \sqrt{2}$
  
  Explanation:
  
  Try to rationalize the denominator. Multiply numerator and denominatr by $\left(\sqrt{4} - \sqrt{2}\right)$
  
  $\sqrt{16} \frac{\sqrt{4} - \sqrt{2}}{\left(\sqrt{4} + \sqrt{2}\right) \cdot \left(\sqrt{4} - \sqrt{2}\right)}$
  
  $4 \cdot \frac{2 - \sqrt{2}}{4 - 2}$
  
  $4 \cdot \frac{2 - \sqrt{2}}{2}$
  
  $2 \cdot \left(2 - \sqrt{2}\right)$
  
  $4 - 2 \sqrt{2}$
  
  Multiply through by $\frac{2 - \sqrt{2}}{2 - \sqrt{2}}$ and work through to get $4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$
  
  Explanation:
  
  $\frac{\sqrt{16}}{\sqrt{4} + \sqrt{2}}$
  
  Let's first take the square roots of the perfect squares:
  
  $\frac{4}{2 + \sqrt{2}}$
  
  In order to simplify, we need the square root out from the denominator. The way to do this is to ensure that when we do FOIL (the process of multiplying 2 quantities within brackets), we don't end up with more square roots. To do that, we'll multiply by $\left(2 - \sqrt{2}\right)$ which will eliminate that possibility (like this):
  
  $\frac{4}{2 + \sqrt{2}} \cdot \left(\frac{2 - \sqrt{2}}{2 - \sqrt{2}}\right)$
  
  $\frac{4 \cdot 2 - 4 \sqrt{2}}{2 \cdot 2 - 2 \sqrt{2} + 2 \sqrt{2} - \sqrt{2} \sqrt{2}}$
  
  $\frac{8 - 4 \sqrt{2}}{4 - 2} = \frac{8 - 4 \sqrt{2}}{2} = 4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$
explanation: The document contains a math question and a detailed answer to the question. The question is about simplifying the expression \(\sqrt{16} / (\sqrt{4} + \sqrt{2})\), and the answer provides a step-by-step solution to simplify the expression to \(4 - 2\sqrt{2}\).
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 do you simplify sqrt (16) / (sqrt (4) + sqrt (2))?
  
  It is
  
  sqrt (16) / (sqrt (4) + sqrt (2))=4/[sqrt2(sqrt2+1)]= 2sqrt2/(sqrt2+1)=2*sqrt2(sqrt2-1)/[(sqrt2+1)*(sqrt2-1)]= 2sqrt2(sqrt2-1)
  
  Jun 26, 2016
  
  $4 - 2 \sqrt{2}$
  
  Explanation:
  
  Try to rationalize the denominator. Multiply numerator and denominatr by $\left(\sqrt{4} - \sqrt{2}\right)$
  
  $\sqrt{16} \frac{\sqrt{4} - \sqrt{2}}{\left(\sqrt{4} + \sqrt{2}\right) \cdot \left(\sqrt{4} - \sqrt{2}\right)}$
  
  $4 \cdot \frac{2 - \sqrt{2}}{4 - 2}$
  
  $4 \cdot \frac{2 - \sqrt{2}}{2}$
  
  $2 \cdot \left(2 - \sqrt{2}\right)$
  
  $4 - 2 \sqrt{2}$
  
  Multiply through by $\frac{2 - \sqrt{2}}{2 - \sqrt{2}}$ and work through to get $4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$
  
  Explanation:
  
  $\frac{\sqrt{16}}{\sqrt{4} + \sqrt{2}}$
  
  Let's first take the square roots of the perfect squares:
  
  $\frac{4}{2 + \sqrt{2}}$
  
  In order to simplify, we need the square root out from the denominator. The way to do this is to ensure that when we do FOIL (the process of multiplying 2 quantities within brackets), we don't end up with more square roots. To do that, we'll multiply by $\left(2 - \sqrt{2}\right)$ which will eliminate that possibility (like this):
  
  $\frac{4}{2 + \sqrt{2}} \cdot \left(\frac{2 - \sqrt{2}}{2 - \sqrt{2}}\right)$
  
  $\frac{4 \cdot 2 - 4 \sqrt{2}}{2 \cdot 2 - 2 \sqrt{2} + 2 \sqrt{2} - \sqrt{2} \sqrt{2}}$
  
  $\frac{8 - 4 \sqrt{2}}{4 - 2} = \frac{8 - 4 \sqrt{2}}{2} = 4 - 2 \sqrt{2} = 2 \left(2 - \sqrt{2}\right)$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer to the question. The question is about simplifying the expression \(\sqrt{16} / (\sqrt{4} + \sqrt{2})\), and the answer provides a step-by-step solution to simplify the expression to \(4 - 2\sqrt{2}\).
  
  Label: question-and-answer