Is the square root of 77 an irrational number?

Answer 1

77 is not a perfect square.

Another way to look at it is this (use a calculator)

sqrt(77) ~= 8.774964387

Notice how there is no point where the numbers begin to repeat themselves (e.g., 1.222322232223)

Both reasons tell us that sqrt(77) is an irrational number.

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Yes - the square roots of all non-perfect square integers (ie the numbers which are 1², 2², 3², ...) are irrational.