shaniehh
  • shaniehh
A student is attempting to show that the product of a nonzero rational number and an irrational number is always rational or always irrational.
Algebra
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SOLVED
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katieb
  • katieb
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shaniehh
  • shaniehh
Part A: Find the value of \[0.22 \] * \[\sqrt{112}\]and place it in simplified form.
shaniehh
  • shaniehh
Part B: Is the answer in Part A irrational or rational? Make a conjecture about the product of a nonzero rational number and an irrational number.
anonymous
  • anonymous
not really clear what the question is find the value of what?

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shaniehh
  • shaniehh
I need help with part B the answer to A is 2.32826115
anonymous
  • anonymous
it just says "make a conjecture" but the truth is that a rational number times an irrational number is always irrational
anonymous
  • anonymous
the proof is very simple if you want to see it
shaniehh
  • shaniehh
yes please
anonymous
  • anonymous
a rational number is any number that can be expressed as a fraction (ratio of two integers) an irrational number is one that cannot be expressed that way we need to know this to start
anonymous
  • anonymous
perhaps the easiest way to show that the product of an irrational number and a rational number is irrational is to prove it by contradiction, that is, assume that the product IS rational, then get a contradiction
anonymous
  • anonymous
so suppose \(a\) is rational, \(b\) is irrational and \(ab=c\) is rational that means that \(b=\frac{c}{a}\) is irrational, but since both \(c\) and \(a\) are rational so is \(\frac{c}{a}\) contradicting the fact that \(b\) if irrational
anonymous
  • anonymous
btw if this seems like kicking the can down the road, it is not you can prove for yourself that if \(c, a\) are rational, then so is \(\frac{c}{a}\) by writing them both as the ratio of two integers, invert and multiply to get another ratio of two integers
shaniehh
  • shaniehh
thank you

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