Question

determine if number is rational / irrational
question
determine if \\(\sqrt{57}\\) is rational or irrational and give a reason for your answer.
answer
attempt 1 out of 2
the number \\(\sqrt{57}\\) is
because

Explanation:

Step1: Recall Rational Number Definition

A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$. For square roots, $\sqrt{n}$ is rational iff $n$ is a perfect square.

Step2: Check if 57 is a Perfect Square

Find integers $a$ such that $a^2 = 57$. We know $7^2 = 49$ and $8^2 = 64$. Since $49 < 57 < 64$, there's no integer $a$ with $a^2 = 57$. Thus, $\sqrt{57}$ cannot be expressed as a fraction of integers.

Answer:

The number $\sqrt{57}$ is irrational because 57 is not a perfect square, so $\sqrt{57}$ cannot be expressed as a ratio of two integers.