Question

√6, -8, 3/5, √9; columns: rational, irrational; for -8, irrational is boxed (maybe a question about classifying numbers as rational or irrational)

Explanation:

Step1: Recall Rational/Irrational Definitions

A rational number can be expressed as $\frac{p}{q}$ ($p,q$ integers, $q
eq0$), or is a terminating/repeating decimal, or an integer. An irrational number is non - repeating, non - terminating, and cannot be expressed as a fraction of two integers.

Step2: Analyze $\boldsymbol{\sqrt{6}}$

To check if $\sqrt{6}$ is rational or irrational, we assume $\sqrt{6}=\frac{p}{q}$ (in lowest terms, $p,q$ integers, $q
eq0$). Then $6=\frac{p^{2}}{q^{2}}$, or $p^{2} = 6q^{2}$. So $p^{2}$ is divisible by 6, which means $p$ is divisible by 6 (since 6 is not a perfect square). Let $p = 6k$ for some integer $k$. Then $(6k)^{2}=6q^{2}$, $36k^{2}=6q^{2}$, $6k^{2}=q^{2}$. So $q^{2}$ is divisible by 6, and $q$ is divisible by 6. But this contradicts the assumption that $\frac{p}{q}$ is in lowest terms. So $\sqrt{6}$ is irrational.

Step3: Analyze $\boldsymbol{-8}$

-8 is an integer. Integers are rational numbers because we can write $-8=\frac{-8}{1}$, where $-8$ and $1$ are integers and $1
eq0$. So -8 is rational, so the "Irrational" box for -8 should be unmarked (or we should note that -8 is rational).

Step4: Analyze $\boldsymbol{\frac{3}{5}}$

$\frac{3}{5}$ is a fraction of two integers (3 and 5, with 5≠0). By the definition of rational numbers, $\frac{3}{5}$ is rational.

Step5: Analyze $\boldsymbol{\sqrt{9}}$

We know that $\sqrt{9}=3$ (since $3\times3 = 9$). 3 is an integer, and integers are rational (we can write $3=\frac{3}{1}$). So $\sqrt{9}$ is rational.

Answer:

• For $\sqrt{6}$: Irrational (mark the "Irrational" column)
• For $-8$: Rational (unmark the "Irrational" column, since -8 is rational)
• For $\frac{3}{5}$: Rational (mark the "Rational" column)
• For $\sqrt{9}$: Rational (mark the "Rational" column)

If we are correcting the table:

Number	Rational	Irrational
$-8$	$\boldsymbol{\checkmark}$
$\frac{3}{5}$	$\boldsymbol{\checkmark}$
$\sqrt{9}$	$\boldsymbol{\checkmark}$