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Answer :
Sure, I'd be happy to help! Let's determine which numbers from the list are rational numbers.
### What is a Rational Number?
A rational number is any number that can be expressed as the quotient or fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Rational numbers can also include integers and fractions with repeating or terminating decimals.
### List of Given Numbers:
1. [tex]\( \frac{5}{4} \)[/tex]
2. 15.090090009...
3. [tex]\( \sqrt{0.09} \)[/tex]
4. [tex]\( 12.9\overline{15} \)[/tex]
5. [tex]\(-89\)[/tex]
6. 3.14159...
7. 0.9375
8. [tex]\( \sqrt{43} \)[/tex]
### Analyzing Each Number:
1. [tex]\( \frac{5}{4} \)[/tex]:
- This is clearly a rational number, as it is expressed as a fraction.
2. 15.090090009...:
- This number does not have a pattern in its decimal expansion, and it doesn’t terminate or repeat. Hence, it is not a rational number.
3. [tex]\( \sqrt{0.09} \)[/tex]:
- Calculate [tex]\( \sqrt{0.09} \)[/tex]. It equals [tex]\( 0.3 \)[/tex], which can be written as [tex]\( \frac{3}{10} \)[/tex]. Therefore, it is a rational number.
4. [tex]\( 12.9\overline{15} \)[/tex]:
- This number is a repeating decimal, where "15" repeats. Hence, it can be expressed as a fraction and is a rational number.
5. [tex]\(-89\)[/tex]:
- An integer is a rational number because it can be expressed as [tex]\( \frac{-89}{1} \)[/tex].
6. 3.14159...:
- This is a non-repeating, non-terminating decimal, commonly known as an approximation of pi ([tex]\(\pi\)[/tex]), which is not rational.
7. 0.9375:
- This is a terminating decimal and can be expressed as [tex]\( \frac{9375}{10000} \)[/tex]. Therefore, it is a rational number.
8. [tex]\( \sqrt{43} \)[/tex]:
- Since 43 is not a perfect square, [tex]\( \sqrt{43} \)[/tex] is an irrational number.
### Conclusion:
The rational numbers from the list are:
- [tex]\( \frac{5}{4} \)[/tex]
- [tex]\( \sqrt{0.09} \)[/tex] (equal to 0.3)
- [tex]\( 12.9\overline{15} \)[/tex]
- [tex]\(-89\)[/tex]
- 0.9375
These numbers can all be expressed as fractions of integers.
### What is a Rational Number?
A rational number is any number that can be expressed as the quotient or fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Rational numbers can also include integers and fractions with repeating or terminating decimals.
### List of Given Numbers:
1. [tex]\( \frac{5}{4} \)[/tex]
2. 15.090090009...
3. [tex]\( \sqrt{0.09} \)[/tex]
4. [tex]\( 12.9\overline{15} \)[/tex]
5. [tex]\(-89\)[/tex]
6. 3.14159...
7. 0.9375
8. [tex]\( \sqrt{43} \)[/tex]
### Analyzing Each Number:
1. [tex]\( \frac{5}{4} \)[/tex]:
- This is clearly a rational number, as it is expressed as a fraction.
2. 15.090090009...:
- This number does not have a pattern in its decimal expansion, and it doesn’t terminate or repeat. Hence, it is not a rational number.
3. [tex]\( \sqrt{0.09} \)[/tex]:
- Calculate [tex]\( \sqrt{0.09} \)[/tex]. It equals [tex]\( 0.3 \)[/tex], which can be written as [tex]\( \frac{3}{10} \)[/tex]. Therefore, it is a rational number.
4. [tex]\( 12.9\overline{15} \)[/tex]:
- This number is a repeating decimal, where "15" repeats. Hence, it can be expressed as a fraction and is a rational number.
5. [tex]\(-89\)[/tex]:
- An integer is a rational number because it can be expressed as [tex]\( \frac{-89}{1} \)[/tex].
6. 3.14159...:
- This is a non-repeating, non-terminating decimal, commonly known as an approximation of pi ([tex]\(\pi\)[/tex]), which is not rational.
7. 0.9375:
- This is a terminating decimal and can be expressed as [tex]\( \frac{9375}{10000} \)[/tex]. Therefore, it is a rational number.
8. [tex]\( \sqrt{43} \)[/tex]:
- Since 43 is not a perfect square, [tex]\( \sqrt{43} \)[/tex] is an irrational number.
### Conclusion:
The rational numbers from the list are:
- [tex]\( \frac{5}{4} \)[/tex]
- [tex]\( \sqrt{0.09} \)[/tex] (equal to 0.3)
- [tex]\( 12.9\overline{15} \)[/tex]
- [tex]\(-89\)[/tex]
- 0.9375
These numbers can all be expressed as fractions of integers.
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