Question

Determine if square root of 75 is rational or irrational and give a reason for your answer.

Answer 1

To determine whether the square root of 75 is rational or irrational, we can start by simplifying the square root.

First, we can factor 75 into its prime factors:

\[
75 = 25 \times 3 = 5^2 \times 3
\]

From this factorization, we can express the square root of 75 as follows:

\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5 \sqrt{3}
\]

Now we need to consider whether \( 5 \sqrt{3} \) is rational or irrational. The number \( 5 \) is rational, but \( \sqrt{3} \) is known to be irrational (since 3 is not a perfect square).

The product of a rational number (5) and an irrational number (\( \sqrt{3} \)) is always irrational.

Thus, we conclude that \( \sqrt{75} = 5 \sqrt{3} \) is irrational.

In summary, the square root of 75 is **irrational** because it can be expressed as the product of a rational number and an irrational number.