Solved on Dec 05, 2023

Classify numbers as rational or irrational: $-28.\overline{17}$ , $-\sqrt{16}$ , $\frac{6}{19}$ , $5\pi$ , $\sqrt{22}$ .

STEP 1

Assumptions
1. A rational number can be expressed as the quotient of two integers (a fraction) where the denominator is not zero.
2. An irrational number cannot be expressed as a simple fraction; it's decimal goes on forever without repeating.
3. A repeating decimal is a rational number.
4. The square root of a non-perfect square is an irrational number.
5. The square root of a perfect square is a rational number.
6. $\pi$ is an irrational number.

STEP 2

We will classify $-28.\overline{17}$ by determining if it can be expressed as a fraction.
Since $-28.\overline{17}$ is a repeating decimal, it can be expressed as a fraction. Therefore, it is a rational number.

STEP 3

We will classify $-\sqrt{16}$ by determining if it can be expressed as a fraction.
Since $-\sqrt{16} = -4$ and $-4$ can be expressed as a fraction $-\frac{4}{1}$ , it is a rational number.

STEP 4

We will classify $\frac{6}{19}$ by determining if it can be expressed as a fraction.
Since $\frac{6}{19}$ is already in fraction form and the denominator is not zero, it is a rational number.

STEP 5

We will classify $5\pi$ by determining if it can be expressed as a fraction.
Since $\pi$ is an irrational number and multiplying by a rational number ( $5$ in this case) does not change this property, $5\pi$ is an irrational number.

STEP 6

We will classify $\sqrt{22}$ by determining if it can be expressed as a fraction.
Since $22$ is not a perfect square, $\sqrt{22}$ cannot be expressed as a simple fraction and is therefore an irrational number.

STEP 7

Now we will fill in the table with the classifications determined in the previous steps.
\begin{tabular}{|c|c|c|} \hline & rational & irrational \\ \hline $-28 . \overline{17}$ & X & 0 \\ \hline $-\sqrt{16}$ & X & 0 \\ \hline $\frac{6}{19}$ & X & 0 \\ \hline $5 \pi$ & 0 & X \\ \hline $\sqrt{22}$ & 0 & X \\ \hline \end{tabular}

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Classify numbers as rational or irrational: $-28.\overline{17}$ , $-\sqrt{16}$ , $\frac{6}{19}$ , $5\pi$ , $\sqrt{22}$ .

STEP 1

STEP 2

We will classify $-28.\overline{17}$ by determining if it can be expressed as a fraction.
Since $-28.\overline{17}$ is a repeating decimal, it can be expressed as a fraction. Therefore, it is a rational number.

STEP 3

We will classify $-\sqrt{16}$ by determining if it can be expressed as a fraction.
Since $-\sqrt{16} = -4$ and $-4$ can be expressed as a fraction $-\frac{4}{1}$ , it is a rational number.

STEP 4

We will classify $\frac{6}{19}$ by determining if it can be expressed as a fraction.
Since $\frac{6}{19}$ is already in fraction form and the denominator is not zero, it is a rational number.

STEP 5

We will classify $5\pi$ by determining if it can be expressed as a fraction.
Since $\pi$ is an irrational number and multiplying by a rational number ( $5$ in this case) does not change this property, $5\pi$ is an irrational number.

STEP 6

We will classify $\sqrt{22}$ by determining if it can be expressed as a fraction.
Since $22$ is not a perfect square, $\sqrt{22}$ cannot be expressed as a simple fraction and is therefore an irrational number.

STEP 7

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Classify numbers as rational or irrational: −28.17‾-28.\overline{17}−28.17, −16-\sqrt{16}−16​, 619\frac{6}{19}196​, 5π5\pi5π, 22\sqrt{22}22​.

STEP 1

STEP 2

We will classify −28.17‾-28.\overline{17}−28.17 by determining if it can be expressed as a fraction.Since −28.17‾-28.\overline{17}−28.17 is a repeating decimal, it can be expressed as a fraction. Therefore, it is a rational number.

STEP 3

We will classify −16-\sqrt{16}−16​ by determining if it can be expressed as a fraction.Since −16=−4-\sqrt{16} = -4−16​=−4 and −4-4−4 can be expressed as a fraction −41-\frac{4}{1}−14​, it is a rational number.

STEP 4

We will classify 619\frac{6}{19}196​ by determining if it can be expressed as a fraction.Since 619\frac{6}{19}196​ is already in fraction form and the denominator is not zero, it is a rational number.

STEP 5

We will classify 5π5\pi5π by determining if it can be expressed as a fraction.Since π\piπ is an irrational number and multiplying by a rational number (555 in this case) does not change this property, 5π5\pi5π is an irrational number.

STEP 6

We will classify 22\sqrt{22}22​ by determining if it can be expressed as a fraction.Since 222222 is not a perfect square, 22\sqrt{22}22​ cannot be expressed as a simple fraction and is therefore an irrational number.

STEP 7

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Classify numbers as rational or irrational: $-28.\overline{17}$ , $-\sqrt{16}$ , $\frac{6}{19}$ , $5\pi$ , $\sqrt{22}$ .

We will classify $-28.\overline{17}$ by determining if it can be expressed as a fraction.
Since $-28.\overline{17}$ is a repeating decimal, it can be expressed as a fraction. Therefore, it is a rational number.

We will classify $-\sqrt{16}$ by determining if it can be expressed as a fraction.
Since $-\sqrt{16} = -4$ and $-4$ can be expressed as a fraction $-\frac{4}{1}$ , it is a rational number.

We will classify $\frac{6}{19}$ by determining if it can be expressed as a fraction.
Since $\frac{6}{19}$ is already in fraction form and the denominator is not zero, it is a rational number.

We will classify $5\pi$ by determining if it can be expressed as a fraction.
Since $\pi$ is an irrational number and multiplying by a rational number ( $5$ in this case) does not change this property, $5\pi$ is an irrational number.

We will classify $\sqrt{22}$ by determining if it can be expressed as a fraction.
Since $22$ is not a perfect square, $\sqrt{22}$ cannot be expressed as a simple fraction and is therefore an irrational number.