Question

determine whether each number is rational or irrational.
chart with columns: (blank), rational, irrational. numbers listed: $-\frac{3}{7}$, $\frac{\sqrt{99}}{3}$, $\sqrt{37}$, $0.\overline{82}$

Explanation:

For $-\frac{3}{7}$:

Step1: Recall rational number definition

A rational number is a number that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q
eq0$.
$-\frac{3}{7}$ is in the form $\frac{p}{q}$ with $p = - 3$, $q=7$ (integers, $q
eq0$).

Step2: Conclude type

So $-\frac{3}{7}$ is rational.

For $\frac{\sqrt{99}}{3}$:

Step1: Simplify $\sqrt{99}$

$\sqrt{99}=\sqrt{9\times11} = 3\sqrt{11}$, so $\frac{\sqrt{99}}{3}=\frac{3\sqrt{11}}{3}=\sqrt{11}$.

Step2: Recall irrational number definition

An irrational number is a number that cannot be expressed as $\frac{p}{q}$ ( $p,q$ integers, $q
eq0$) and has a non - repeating, non - terminating decimal expansion. $\sqrt{11}$ is irrational (since 11 is not a perfect square), so $\frac{\sqrt{99}}{3}$ is irrational.

For $\sqrt{37}$:

Step1: Check if 37 is a perfect square

The perfect squares around 37 are $5^2 = 25$ and $6^2=36$, $7^2 = 49$. Since 37 is not a perfect square, $\sqrt{37}$ has a non - repeating, non - terminating decimal expansion.

Step2: Conclude type

By the definition of irrational numbers, $\sqrt{37}$ is irrational.

For $0.\overline{82}$:

Answer:

• $-\frac{3}{7}$: Rational (select the circle under "Rational" for $-\frac{3}{7}$)
• $\frac{\sqrt{99}}{3}$: Irrational (select the circle under "Irrational" for $\frac{\sqrt{99}}{3}$)
• $\sqrt{37}$: Irrational (select the circle under "Irrational" for $\sqrt{37}$)
• $0.\overline{82}$: Rational (select the circle under "Rational" for $0.\overline{82}$)