Determine if square root of, 53, end square root

Question

53
​
is rational or irrational and give a reason for your answer.

Answer 1

The square root of 53, denoted as \( \sqrt{53} \), is an irrational number.

To determine whether a square root is rational or irrational, we can check if the number under the square root is a perfect square. A rational number can be expressed as the ratio of two integers (i.e., in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)).

53 is not a perfect square because there are no two integers that, when multiplied together, equal 53. The nearest perfect squares are \( 49 \) (which is \( 7^2 \)) and \( 64 \) (which is \( 8^2 \)). Since \( 7^2 < 53 < 8^2 \), we conclude that \( \sqrt{53} \) cannot be expressed as a ratio of integers.

Thus, \( \sqrt{53} \) is irrational.