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sqrt(8) = sqrt(4*2) = 2*sqrt(2).
Even without given that sqrt(2) is a rational, you can give that the square root of 2 starts converging onto the "Pythagoras Constant" eventually, as it takes an infinite amount of digits to square root an integer that is not perfectly squared.
Thus, an rational x irrational = irrational, thus the sqrt(8) is irrational (an approximation is 2.8284271247...).
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