Determine if square root of, 57, end square root

Question

57
​
is rational or irrational and give a reason for your answer.

Answer 1

To determine whether \( \sqrt{57} \) is rational or irrational, we can analyze the number 57. A rational number can be expressed as a fraction of two integers (where the denominator is not zero).

To ascertain whether \( \sqrt{57} \) is rational, we need to check if 57 is a perfect square:

• The perfect squares around 57 are \( 49 (7^2) \) and \( 64 (8^2) \). Since 57 is not equal to either 49 or 64, it is not a perfect square.

Moreover, since 57 is not a product of a pair of identical integers, we conclude that \( \sqrt{57} \) cannot be expressed as a fraction of two integers.

Thus, \( \sqrt{57} \) is irrational.

Reason: 57 is not a perfect square, confirming that its square root cannot be expressed as a simple fraction.