Question

classify each number below as a rational number or an irrational number.
\\(\frac{12}{7}\\)
\\(-14\pi\\)
\\(17.64\\)
\\(-\sqrt{9}\\)
\\(-\sqrt{11}\\)

Explanation:

Step1: Classify $\frac{12}{7}$

Rational numbers are ratios of integers. $\frac{12}{7}$ is $\frac{\text{integer}}{\text{non-zero integer}}$, so it is rational.

Step2: Classify $-14\pi$

$\pi$ is irrational; multiplying by integer $-14$ keeps it irrational. So $-14\pi$ is irrational.

Step3: Classify $17.\overline{64}$

Repeating decimals are rational (can be written as $\frac{1747}{99}$). So $17.\overline{64}$ is rational.

Step4: Classify $-\sqrt{9}$

$\sqrt{9}=3$, so $-\sqrt{9}=-3$, an integer (subset of rational numbers). So $-\sqrt{9}$ is rational.

Step5: Classify $-\sqrt{13}$

13 is not a perfect square, so $\sqrt{13}$ is irrational; negating it keeps it irrational. So $-\sqrt{13}$ is irrational.

Answer:

Number	Rational	Irrational
$-14\pi$		$\checkmark$
$17.\overline{64}$	$\checkmark$
$-\sqrt{9}$	$\checkmark$
$-\sqrt{13}$		$\checkmark$