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Answer :
Sure! Let's determine which numbers from the list are rational. A rational number is any number that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, and [tex]\(b\)[/tex] is not zero.
Let's examine each number:
1. -6: This is an integer, which is a type of rational number because it can be expressed as [tex]\(-6/1\)[/tex]. So, -6 is rational.
2. 0: This is also an integer and can be expressed as [tex]\(0/1\)[/tex]. Thus, 0 is a rational number.
3. 76: This is an integer and can be written as [tex]\(76/1\)[/tex]. So, 76 is rational.
4. 5.17: This is a terminating decimal, which can be written as a fraction: [tex]\(517/100\)[/tex]. Therefore, 5.17 is rational.
5. [tex]\(\sqrt{2}\)[/tex]: The square root of 2 is not an integer and cannot be expressed as a fraction of two integers. It is an irrational number.
6. [tex]\(\sqrt{16}\)[/tex]: This equals 4, which is an integer. As it can be written as [tex]\(4/1\)[/tex], it is a rational number.
7. [tex]\(8 \frac{1}{8}\)[/tex]: This is a mixed number, which can be converted to an improper fraction: [tex]\(65/8\)[/tex]. Therefore, it is rational.
8. [tex]\(-\frac{7}{2}\)[/tex]: This is already in the form of a fraction [tex]\(\frac{a}{b}\)[/tex] with integers. So [tex]\(-\frac{7}{2}\)[/tex] is rational.
9. 1.4747747774...: This is a non-terminating, non-repeating decimal which cannot be expressed as a fraction, making it irrational.
In summary, the rational numbers from the list are:
- A. [tex]\(8 \frac{1}{8}\)[/tex]
- B. 0
- E. 5.17
- F. [tex]\(\sqrt{16}\)[/tex]
- G. 76
- H. [tex]\(-\frac{7}{2}\)[/tex]
- I. -6
Let's examine each number:
1. -6: This is an integer, which is a type of rational number because it can be expressed as [tex]\(-6/1\)[/tex]. So, -6 is rational.
2. 0: This is also an integer and can be expressed as [tex]\(0/1\)[/tex]. Thus, 0 is a rational number.
3. 76: This is an integer and can be written as [tex]\(76/1\)[/tex]. So, 76 is rational.
4. 5.17: This is a terminating decimal, which can be written as a fraction: [tex]\(517/100\)[/tex]. Therefore, 5.17 is rational.
5. [tex]\(\sqrt{2}\)[/tex]: The square root of 2 is not an integer and cannot be expressed as a fraction of two integers. It is an irrational number.
6. [tex]\(\sqrt{16}\)[/tex]: This equals 4, which is an integer. As it can be written as [tex]\(4/1\)[/tex], it is a rational number.
7. [tex]\(8 \frac{1}{8}\)[/tex]: This is a mixed number, which can be converted to an improper fraction: [tex]\(65/8\)[/tex]. Therefore, it is rational.
8. [tex]\(-\frac{7}{2}\)[/tex]: This is already in the form of a fraction [tex]\(\frac{a}{b}\)[/tex] with integers. So [tex]\(-\frac{7}{2}\)[/tex] is rational.
9. 1.4747747774...: This is a non-terminating, non-repeating decimal which cannot be expressed as a fraction, making it irrational.
In summary, the rational numbers from the list are:
- A. [tex]\(8 \frac{1}{8}\)[/tex]
- B. 0
- E. 5.17
- F. [tex]\(\sqrt{16}\)[/tex]
- G. 76
- H. [tex]\(-\frac{7}{2}\)[/tex]
- I. -6
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