Answer: Irrational
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Step-by-step explanation:
√2 is irrational because we cannot write it as a fraction, or ratio, of two integers. The proof of this is fairly lengthy so I recommend searching out "square root of 2 irrational proof", or something along those lines.
Since √2 is irrational, this makes 4√2 irrational as well. The proof will be similar to the previous proof.
4/18 is rational since it is a ratio of two integers.
The product of an irrational value and rational value leads to an irrational answer
The proof of this is given in the next section below.
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Proof by contradiction:
p,q,r,s = four nonzero integers
A = some rational number = p/q
B = some irrational number
C = some other rational number = r/s
Let's assume for a moment that A*B is rational
If that's the case, then,
A*B = C
(p/q)*B = r/s
B = (r/s)*(q/p)
B = (rq)/(sp)
B = (some integer)/(some nonzero integer)
B = some rational number
But we defined B as some irrational number, which contradicts what we arrived at just now. A number can't be both rational and irrational at the same time. The very name "irrational" literally means "not rational".
Therefore, A*B must be irrational.