Looking for reliable answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
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].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
