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Sagot :
Sure! Let's work through each part of the question step by step.
### 1. Determine if each given number is an integer or an irrational number.
#### (a) [tex]\(\sqrt{9} = 3\)[/tex]
- Explanation: [tex]\(\sqrt{9}\)[/tex] means the square root of 9. Since [tex]\(3\)[/tex] is a number such that [tex]\(3^2 = 9\)[/tex], [tex]\(\sqrt{9} = 3\)[/tex]. The number 3 is an integer.
- Conclusion: Integer.
#### (b) [tex]\(\sqrt{19}\)[/tex]
- Explanation: [tex]\(\sqrt{19}\)[/tex] means the square root of 19. Since 19 is not a perfect square, its square root is not an integer. Instead, it is an irrational number, which cannot be expressed as a fraction.
- Conclusion: Irrational number.
#### (c) [tex]\(\sqrt{39} = \sqrt{30} \times \sqrt{9}\)[/tex]
- Explanation: [tex]\(\sqrt{39}\)[/tex] means the square root of 39. Since 39 is not a perfect square, its square root is not an integer. The product [tex]\(\sqrt{30} \times \sqrt{9}\)[/tex] simplifies to [tex]\(\sqrt{270}\)[/tex], which also isn't a perfect square, thus confirming [tex]\(\sqrt{39}\)[/tex] is irrational.
- Conclusion: Irrational number.
#### (d) [tex]\(\sqrt{49} = 7\)[/tex]
- Explanation: [tex]\(\sqrt{49}\)[/tex] means the square root of 49. Since [tex]\(7\)[/tex] is a number such that [tex]\(7^2 = 49\)[/tex], [tex]\(\sqrt{49} = 7\)[/tex]. The number 7 is an integer.
- Conclusion: Integer.
#### (e) [tex]\(\sqrt{99}\)[/tex]
- Explanation: [tex]\(\sqrt{99}\)[/tex] means the square root of 99. Since 99 is not a perfect square, its square root is not an integer. Instead, it is an irrational number.
- Conclusion: Irrational number.
### 2. List the rational numbers in the given list.
Given list: [tex]\[\left\{\sqrt{1},\ 7 \frac{5}{12},\ -38,\ \sqrt{160},\ -\sqrt{2.25},\ -\sqrt{35}\right\}\][/tex]
- [tex]\(\sqrt{1}\)[/tex]: [tex]\(\sqrt{1} = 1.0\)[/tex]. Since 1.0 can be expressed as the fraction 1/1, it is a rational number.
- [tex]\(7\frac{5}{12}\)[/tex]: This is a mixed number that can be converted to an improper fraction [tex]\( \frac{89}{12} \)[/tex], making it a rational number.
- [tex]\(-38\)[/tex]: This is an integer, and all integers are rational numbers.
- [tex]\(\sqrt{160}\)[/tex]: [tex]\(\sqrt{160}\)[/tex] is not an integer and cannot be expressed as a fraction. Therefore, it is an irrational number.
- [tex]\(-\sqrt{2.25}\)[/tex]: [tex]\(\sqrt{2.25} = 1.5\)[/tex], so [tex]\(-\sqrt{2.25} = -1.5\)[/tex]. Since -1.5 can be expressed as [tex]\(\frac{-3}{2}\)[/tex], it is a rational number.
- [tex]\(-\sqrt{35}\)[/tex]: [tex]\(\sqrt{35}\)[/tex] is not an integer and cannot be expressed as a fraction. Therefore, it is an irrational number.
### 3. Writing the rational numbers:
The rational numbers from the given list are:
[tex]\[ \left\{ \sqrt{1} (=1.0),\ -38 \right\} \][/tex]
We can conclude and write:
- 1.0 (which was [tex]\(\sqrt{1}\)[/tex])
- -38
Therefore, the rational numbers in the list are: [tex]\(\mathbf{[1.0, -38]}\)[/tex].
### 1. Determine if each given number is an integer or an irrational number.
#### (a) [tex]\(\sqrt{9} = 3\)[/tex]
- Explanation: [tex]\(\sqrt{9}\)[/tex] means the square root of 9. Since [tex]\(3\)[/tex] is a number such that [tex]\(3^2 = 9\)[/tex], [tex]\(\sqrt{9} = 3\)[/tex]. The number 3 is an integer.
- Conclusion: Integer.
#### (b) [tex]\(\sqrt{19}\)[/tex]
- Explanation: [tex]\(\sqrt{19}\)[/tex] means the square root of 19. Since 19 is not a perfect square, its square root is not an integer. Instead, it is an irrational number, which cannot be expressed as a fraction.
- Conclusion: Irrational number.
#### (c) [tex]\(\sqrt{39} = \sqrt{30} \times \sqrt{9}\)[/tex]
- Explanation: [tex]\(\sqrt{39}\)[/tex] means the square root of 39. Since 39 is not a perfect square, its square root is not an integer. The product [tex]\(\sqrt{30} \times \sqrt{9}\)[/tex] simplifies to [tex]\(\sqrt{270}\)[/tex], which also isn't a perfect square, thus confirming [tex]\(\sqrt{39}\)[/tex] is irrational.
- Conclusion: Irrational number.
#### (d) [tex]\(\sqrt{49} = 7\)[/tex]
- Explanation: [tex]\(\sqrt{49}\)[/tex] means the square root of 49. Since [tex]\(7\)[/tex] is a number such that [tex]\(7^2 = 49\)[/tex], [tex]\(\sqrt{49} = 7\)[/tex]. The number 7 is an integer.
- Conclusion: Integer.
#### (e) [tex]\(\sqrt{99}\)[/tex]
- Explanation: [tex]\(\sqrt{99}\)[/tex] means the square root of 99. Since 99 is not a perfect square, its square root is not an integer. Instead, it is an irrational number.
- Conclusion: Irrational number.
### 2. List the rational numbers in the given list.
Given list: [tex]\[\left\{\sqrt{1},\ 7 \frac{5}{12},\ -38,\ \sqrt{160},\ -\sqrt{2.25},\ -\sqrt{35}\right\}\][/tex]
- [tex]\(\sqrt{1}\)[/tex]: [tex]\(\sqrt{1} = 1.0\)[/tex]. Since 1.0 can be expressed as the fraction 1/1, it is a rational number.
- [tex]\(7\frac{5}{12}\)[/tex]: This is a mixed number that can be converted to an improper fraction [tex]\( \frac{89}{12} \)[/tex], making it a rational number.
- [tex]\(-38\)[/tex]: This is an integer, and all integers are rational numbers.
- [tex]\(\sqrt{160}\)[/tex]: [tex]\(\sqrt{160}\)[/tex] is not an integer and cannot be expressed as a fraction. Therefore, it is an irrational number.
- [tex]\(-\sqrt{2.25}\)[/tex]: [tex]\(\sqrt{2.25} = 1.5\)[/tex], so [tex]\(-\sqrt{2.25} = -1.5\)[/tex]. Since -1.5 can be expressed as [tex]\(\frac{-3}{2}\)[/tex], it is a rational number.
- [tex]\(-\sqrt{35}\)[/tex]: [tex]\(\sqrt{35}\)[/tex] is not an integer and cannot be expressed as a fraction. Therefore, it is an irrational number.
### 3. Writing the rational numbers:
The rational numbers from the given list are:
[tex]\[ \left\{ \sqrt{1} (=1.0),\ -38 \right\} \][/tex]
We can conclude and write:
- 1.0 (which was [tex]\(\sqrt{1}\)[/tex])
- -38
Therefore, the rational numbers in the list are: [tex]\(\mathbf{[1.0, -38]}\)[/tex].
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