Answer :
Let's carefully evaluate each expression to determine if it helps to show that the statement "The sum of a rational number and an irrational number is always rational" is false. This will involve checking whether or not the sum is irrational.
1. Expression A: [tex]\(\sqrt{25} + \pi\)[/tex]
- [tex]\(\sqrt{25}\)[/tex] equals 5, which is a rational number.
- [tex]\(\pi\)[/tex] is an irrational number.
- The sum [tex]\(5 + \pi\)[/tex] combines a rational number (5) and an irrational number ([tex]\(\pi\)[/tex]), resulting in an irrational number.
2. Expression B: [tex]\(0.56 + \pi\)[/tex]
- [tex]\(0.56\)[/tex] is a rational number, as it can be expressed as a fraction ([tex]\(\frac{56}{100}\)[/tex]).
- [tex]\(\pi\)[/tex] is an irrational number.
- The sum [tex]\(0.56 + \pi\)[/tex] involves a rational number (0.56) and an irrational number ([tex]\(\pi\)[/tex]), resulting in an irrational number.
3. Expression C: [tex]\(\frac{7}{8} + \sqrt{13}\)[/tex]
- [tex]\(\frac{7}{8}\)[/tex] is a rational number.
- [tex]\(\sqrt{13}\)[/tex] is an irrational number.
- The sum [tex]\(\frac{7}{8} + \sqrt{13}\)[/tex] involves a rational number ([tex]\(\frac{7}{8}\)[/tex]) and an irrational number ([tex]\(\sqrt{13}\)[/tex]), resulting in an irrational number.
4. Expression D: [tex]\(\pi + \sqrt{17}\)[/tex]
- [tex]\(\pi\)[/tex] is an irrational number.
- [tex]\(\sqrt{17}\)[/tex] is an irrational number.
- While both components are irrational, the focus here should be that the resulting sum remains irrational, irrespective of our concern about irrational and rational number addition.
5. Expression E: [tex]\(0.45 + 0.96\)[/tex]
- [tex]\(0.45\)[/tex] is a rational number.
- [tex]\(0.96\)[/tex] is a rational number.
- The sum [tex]\(0.45 + 0.96\)[/tex] combines two rational numbers, resulting in a rational number.
6. Expression F: [tex]\(\sqrt{18} + \sqrt{21}\)[/tex]
- [tex]\(\sqrt{18}\)[/tex] is an irrational number.
- [tex]\(\sqrt{21}\)[/tex] is an irrational number.
- Similar to Expression D, the sum remains irrational but involves two irrational numbers, which is not our current focus.
We are primarily interested in expressions that combine rational and irrational numbers to show that their sum results in an irrational number, thereby falsifying the statement in question.
The expressions that show the given statement is false are:
- Expression A: [tex]\(\sqrt{25} + \pi\)[/tex]
- Expression B: [tex]\(0.56 + \pi\)[/tex]
- Expression C: [tex]\(\frac{7}{8} + \sqrt{13}\)[/tex]
These expressions demonstrate that the sum of a rational number and an irrational number is irrational, thus proving the given statement false.
1. Expression A: [tex]\(\sqrt{25} + \pi\)[/tex]
- [tex]\(\sqrt{25}\)[/tex] equals 5, which is a rational number.
- [tex]\(\pi\)[/tex] is an irrational number.
- The sum [tex]\(5 + \pi\)[/tex] combines a rational number (5) and an irrational number ([tex]\(\pi\)[/tex]), resulting in an irrational number.
2. Expression B: [tex]\(0.56 + \pi\)[/tex]
- [tex]\(0.56\)[/tex] is a rational number, as it can be expressed as a fraction ([tex]\(\frac{56}{100}\)[/tex]).
- [tex]\(\pi\)[/tex] is an irrational number.
- The sum [tex]\(0.56 + \pi\)[/tex] involves a rational number (0.56) and an irrational number ([tex]\(\pi\)[/tex]), resulting in an irrational number.
3. Expression C: [tex]\(\frac{7}{8} + \sqrt{13}\)[/tex]
- [tex]\(\frac{7}{8}\)[/tex] is a rational number.
- [tex]\(\sqrt{13}\)[/tex] is an irrational number.
- The sum [tex]\(\frac{7}{8} + \sqrt{13}\)[/tex] involves a rational number ([tex]\(\frac{7}{8}\)[/tex]) and an irrational number ([tex]\(\sqrt{13}\)[/tex]), resulting in an irrational number.
4. Expression D: [tex]\(\pi + \sqrt{17}\)[/tex]
- [tex]\(\pi\)[/tex] is an irrational number.
- [tex]\(\sqrt{17}\)[/tex] is an irrational number.
- While both components are irrational, the focus here should be that the resulting sum remains irrational, irrespective of our concern about irrational and rational number addition.
5. Expression E: [tex]\(0.45 + 0.96\)[/tex]
- [tex]\(0.45\)[/tex] is a rational number.
- [tex]\(0.96\)[/tex] is a rational number.
- The sum [tex]\(0.45 + 0.96\)[/tex] combines two rational numbers, resulting in a rational number.
6. Expression F: [tex]\(\sqrt{18} + \sqrt{21}\)[/tex]
- [tex]\(\sqrt{18}\)[/tex] is an irrational number.
- [tex]\(\sqrt{21}\)[/tex] is an irrational number.
- Similar to Expression D, the sum remains irrational but involves two irrational numbers, which is not our current focus.
We are primarily interested in expressions that combine rational and irrational numbers to show that their sum results in an irrational number, thereby falsifying the statement in question.
The expressions that show the given statement is false are:
- Expression A: [tex]\(\sqrt{25} + \pi\)[/tex]
- Expression B: [tex]\(0.56 + \pi\)[/tex]
- Expression C: [tex]\(\frac{7}{8} + \sqrt{13}\)[/tex]
These expressions demonstrate that the sum of a rational number and an irrational number is irrational, thus proving the given statement false.
