Question

determine if \\(sqrt{41}\\) is rational or irrational and give a reason for your answer. answer attempt 1 out of 2 the number \\(sqrt{41}\\) is because

Explanation:

Step1: Recall rational number definition

A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$, and $p,q$ have no common factors other than 1 (in lowest terms). For a square root $\sqrt{n}$ to be rational, $n$ must be a perfect square (i.e., there exists an integer $k$ such that $n = k^2$).

Step2: Check if 41 is a perfect square

Find integers $k$ such that $k^2 = 41$. We know that $6^2 = 36$ and $7^2 = 49$. Since $36<41<49$, there is no integer $k$ where $k^2 = 41$. So 41 is not a perfect square.

Step3: Conclude about $\sqrt{41}$

Since 41 is not a perfect square, $\sqrt{41}$ cannot be expressed as a ratio of two integers (it is a non - repeating, non - terminating decimal). So $\sqrt{41}$ is irrational.

Answer:

The number $\sqrt{41}$ is irrational because 41 is not a perfect square, and a square root of a non - perfect square positive integer is irrational.