Question

which number is rational?
a. \\(\sqrt{27}\\)
b. \\(\sqrt{50}\\)
c. \\(\sqrt{64}\\)
d. \\(\sqrt{72}\\)

Explanation:

Step1: Recall the definition of a rational number

A rational number is a number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$. In the case of square roots, a square root of a number is rational if the number is a perfect square (i.e., the square root results in an integer).

Step2: Analyze Option A: $\sqrt{27}$

Simplify $\sqrt{27}$. We know that $27 = 9\times3$, so $\sqrt{27}=\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}$. Since $\sqrt{3}$ is an irrational number (it cannot be expressed as a fraction of two integers), $3\sqrt{3}$ is also irrational.

Step3: Analyze Option B: $\sqrt{50}$

Simplify $\sqrt{50}$. We can write $50 = 25\times2$, so $\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}$. Since $\sqrt{2}$ is irrational, $5\sqrt{2}$ is also irrational.

Step4: Analyze Option C: $\sqrt{64}$

Simplify $\sqrt{64}$. We know that $8\times8 = 64$, so $\sqrt{64}=8$. The number $8$ can be expressed as $\frac{8}{1}$, where $8$ and $1$ are integers and $1
eq0$. So, $8$ (or $\sqrt{64}$) is a rational number.

Step5: Analyze Option D: $\sqrt{72}$

Simplify $\sqrt{72}$. We can write $72 = 36\times2$, so $\sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2}$. Since $\sqrt{2}$ is irrational, $6\sqrt{2}$ is also irrational.

Answer:

C. $\boldsymbol{\sqrt{64}}$