prove that √2-8 is irrational
Answers
It’s irrational. A justification of this fact could go something like this:
First of all, we know 2–√ is irrational.
So, let’s pretend for a moment that 2√8 is rational. Then we know there has to be some rational number q so that 2√8=q .
If we multiply both sides by 8 , we get 2–√=8q . Notice that 8 and q are both rational, so their product must be rational.
[side note: This is easy to prove, if we have a pair of rational numbers they can be written as ab and cd where a , b , c and d are all integers and neither b nor d is zero. But then (ab)(cd) = acbd . We know ac and bd are integers and bd can’t be zero since neither of it’s factors are, so the product of two rational numbers is rational.]
Anyway, if 8q is rational, so is 2–√ . But we know that 2–√ is irrational, which gives us a contradiction. So our assumption that 2√8 is rational must be incorrect. Hence 2√8 is irrational.
Also, the other responders are correct; if you meant 28−−√ , that is rational as it’s equal to 12 .
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Michael Brinkley, former Machinist Trainee at MetalTek (2018-2020)
Answered September 12, 2018
I’m assuming you mean something like square root of 2 divided by 8 or that you want a fraction where 8 is the denominator and square root of 2 is the numerator. Either way, the answer will be the same.
To have it be rational, you need to express both parts as a rational number, or a ratio of two numbers. Since square root of 2 is irrational, putting it over 8 will keep it irrational
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Chris Martin
Answered August 26
Irrational.
The square root of 2 is itself irrational, and you cannot get a rational number by multiplying an irrational number by a rational one (and division is technically multiplication).
The only number you can multiply by the square root of 2 to get a rational product is itself, and that is only because it's the square root of a rational number.
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Robby Goetschalckx, Computer scientist for 11+ years and passionate about math since childhood.
Answered September 12, 2018 · Upvoted by David Joyce, Ph.D. Mathematics, University of Pennsylvania (1979)
It depends on whether you meant 28−−√ or 2√8 . The first one is rational, the second one is irrational.
Your question is ambiguous, so it’s not clear what the answer should be. Please use proper math notation to avoid ambiguities.
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√8 = √2^3, as 8 = 2^3
So 8 = 2√2
Divide this by √2 gives 2.
So the answer is rational.
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