To determine which set contains all rational numbers, let's analyze each set individually:
1. Set: [tex]\(\left\{\frac{1}{3}, -3.45, \sqrt{9}\right\}\)[/tex]
- [tex]\(\frac{1}{3}\)[/tex] is a rational number because it can be expressed as a fraction of two integers.
- [tex]\(-3.45\)[/tex] is a rational number because it can be written as [tex]\(-\frac{345}{100}\)[/tex].
- [tex]\(\sqrt{9}\)[/tex] is equal to 3, which is an integer and hence a rational number.
Therefore, every element in the set [tex]\(\left\{\frac{1}{3}, -3.45, \sqrt{9}\right\}\)[/tex] is rational.
2. Set: [tex]\(\{\pi, 9.25, \sqrt{37}\}\)[/tex]
- [tex]\(\pi\)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers.
- [tex]\(9.25\)[/tex] is a rational number because it can be written as [tex]\(\frac{925}{100}\)[/tex].
- [tex]\(\sqrt{37}\)[/tex] is an irrational number because 37 is not a perfect square.
Therefore, not all elements in the set [tex]\(\{\pi, 9.25, \sqrt{37}\}\)[/tex] are rational.
3. Set: [tex]\(\{3, -9, \sqrt{44}\}\)[/tex]
- [tex]\(3\)[/tex] is an integer and hence a rational number.
- [tex]\(-9\)[/tex] is an integer and hence a rational number.
- [tex]\(\sqrt{44}\)[/tex] is an irrational number because 44 is not a perfect square.
Therefore, not all elements in the set [tex]\(\{3, -9, \sqrt{44}\}\)[/tex] are rational.
4. Set: [tex]\(\{\sqrt{2}, 10, 7\}\)[/tex]
- [tex]\(\sqrt{2}\)[/tex] is an irrational number because 2 is not a perfect square.
- [tex]\(10\)[/tex] is an integer and hence a rational number.
- [tex]\(7\)[/tex] is an integer and hence a rational number.
Therefore, not all elements in the set [tex]\(\{\sqrt{2}, 10, 7\}\)[/tex] are rational.
By analyzing each set, we see that the first set [tex]\(\left\{\frac{1}{3}, -3.45, \sqrt{9}\right\}\)[/tex] contains only rational numbers.
Thus, the number set that contains all rational numbers is:
[tex]\[ \left\{\frac{1}{3}, -3.45, \sqrt{9}\right\} \][/tex]
