Determine if \( \sqrt{75} \) is rational or irrational and give a reason for your answe

To determine if \( \sqrt{75} \) is rational, we can simplify it. Since \( 75 = 25 \times 3 \), we have \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \). Now, the number \( \sqrt{3} \) is known to be irrational (it cannot be expressed as a fraction of two integers). Thus, \( 5\sqrt{3} \) remains irrational because the product of a rational number (5) and an irrational number (\(\sqrt{3}\)) is also irrational. Therefore, \( \sqrt{75} \) is irrational. The square root of a non-perfect square (like 75) always results in an irrational number. Remember, a perfect square is a number that can be expressed as the square of an integer, like 16 or 36. Since 75 does not fall into this category, its square root keeps the irrational quality!