ANSWERS: 1
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Rational numbers always repeat when written as a decimal. The first number you wrote cannot repeat, as each sequence of 7s is unique. Once you see a run of n 7s surrounded by sixes, that never occurs again. 1/sqrt(15) is irrational using reductio ad adsurbum. If it is rational, then sqrt(15) is also rational. If sqrt(15) is rational then it is p/q, a fraction with no common factor. sqrt(15) = p/q 15q^2 = p^2 Therefore p has a factor of 15. It must be 15r for some other integer r. So 15q^2 = 225r^2 q^2 = 15r^2 so q has a factor of 15 too. Contradiction! sqrt(32)/sqrt(2) = sqrt(16) = 4 That's rational.
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