Question

determine if \\(sqrt{53}\\) is rational or irrational and give a reason for your answer. answer attempt 1 out of 2 the number \\(sqrt{53}\\) 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). Also, a rational number's square root is rational only if the number is a perfect square.

Step2: Check if 53 is a perfect square

Find integers $n$ such that $n^2 = 53$. We know $7^2=49$ and $8^2 = 64$. Since $49<53<64$, there's no integer $n$ with $n^2 = 53$. So 53 is not a perfect square.

Step3: Conclude about $\sqrt{53}$

Since 53 is not a perfect square, $\sqrt{53}$ cannot be expressed as a fraction of two integers (because if $\sqrt{53}=\frac{p}{q}$, then $53=\frac{p^2}{q^2}$, so $p^2 = 53q^2$, implying 53 divides $p^2$ and thus 53 divides $p$, let $p = 53k$, then $53q^2=(53k)^2=53^2k^2$, so $q^2 = 53k^2$, implying 53 divides $q$, contradicting $p,q$ having no common factors). So $\sqrt{53}$ is irrational.

Answer:

The number $\sqrt{53}$ is irrational because 53 is not a perfect square (there is no integer $n$ such that $n^2 = 53$), so $\sqrt{53}$ cannot be expressed as a ratio of two integers.