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√15=a/b

(a and b are not common factor).

squaring on both sides, we get,

15= a^2/b^2.

15b^2=a^2 (a=15c)

b^2=15/a^2

(assume, a^2 =225c^2)

15b^2=225c^2

b^2=225c^2/15

b^2=15c^2

c^2=15/b^2

c =15/b.

15 divided both a and b which is a contradiction.

ie, √15 is irrational.