Question

classify each number below as a rational number or an irrational number.

	rational	irrational
$46.83$	$\circ$	$\circ$
$-6\pi$	$\circ$	$\circ$
$2\sqrt{5}$	$\circ$	$\circ$
$92.\overline{15}$	$\circ$	$\circ$

Explanation:

Step1: Analyze $\boldsymbol{\sqrt{9}}$

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. $\sqrt{9} = 3$, and $3=\frac{3}{1}$, so it is rational.

Step2: Analyze $\boldsymbol{46.83}$

$46.83$ is a terminating decimal. Terminating decimals can be written as fractions (e.g., $46.83=\frac{4683}{100}$), so it is rational.

Step3: Analyze $\boldsymbol{-6\pi}$

$\pi$ is an irrational number (it cannot be expressed as a fraction of two integers). Multiplying an irrational number by a non - zero integer ($- 6$ here) still gives an irrational number. So $-6\pi$ is irrational.

Step4: Analyze $\boldsymbol{2\sqrt{5}}$

$\sqrt{5}$ is an irrational number (since 5 is not a perfect square). Multiplying an irrational number by a non - zero integer (2 here) still gives an irrational number. So $2\sqrt{5}$ is irrational.

Step5: Analyze $\boldsymbol{92.\overline{15}}$

A repeating decimal can be expressed as a fraction. Let $x = 92.\overline{15}$, then $100x=9215.\overline{15}$, and $100x - x=9215.\overline{15}-92.\overline{15}$, $99x = 9123$, $x=\frac{9123}{99}$, so it is rational.

Answer:

• $\sqrt{9}$: rational (select the circle under "rational")
• $46.83$: rational (select the circle under "rational")
• $-6\pi$: irrational (select the circle under "irrational")
• $2\sqrt{5}$: irrational (select the circle under "irrational")
• $92.\overline{15}$: rational (select the circle under "rational")