Question
Consider the statement: "The sum of a rational number and an irrational number is always rational."
Choose ,begin emphasis,all,end emphasis, of the expressions that show that this statement is false.
Answer options with 6 options
A.
StartRoot 25 EndRoot plus pi
B.
0.56 plus pi
C.
seven-eighths plus StartRoot 13 EndRoot
D.
pi plus StartRoot 17 EndRoot
E.
0.45 plus 0.96
F.
StartRoot 18 EndRoot plus StartRoot 21 EndRoot
Choose ,begin emphasis,all,end emphasis, of the expressions that show that this statement is false.
Answer options with 6 options
A.
StartRoot 25 EndRoot plus pi
B.
0.56 plus pi
C.
seven-eighths plus StartRoot 13 EndRoot
D.
pi plus StartRoot 17 EndRoot
E.
0.45 plus 0.96
F.
StartRoot 18 EndRoot plus StartRoot 21 EndRoot
Answers
GPT-4o mini
To determine which of the provided expressions demonstrate that the statement "The sum of a rational number and an irrational number is always rational" is false, we are looking for sums that involve a rational number and an irrational number, leading to an irrational result.
A rational number is any number that can be expressed as the quotient of two integers, such as fractions or whole numbers. An irrational number cannot be expressed as a fraction of two integers.
Let's analyze the options:
A. \( \sqrt{25} + \pi \)
- \( \sqrt{25} = 5 \) (rational) + \( \pi \) (irrational) → This is rational + irrational = irrational.
B. \( 0.56 + \pi \)
- \( 0.56 \) (rational) + \( \pi \) (irrational) → This is rational + irrational = irrational.
C. \( \frac{7}{8} + \sqrt{13} \)
- \( \frac{7}{8} \) (rational) + \( \sqrt{13} \) (irrational) → This is rational + irrational = irrational.
D. \( \pi + \sqrt{17} \)
- \( \pi \) (irrational) + \( \sqrt{17} \) (irrational) → This is not a sum of rational + irrational; we can't determine if it's rational or irrational without further information.
E. \( 0.45 + 0.96 \)
- Both \( 0.45 \) and \( 0.96 \) are rational → This is rational + rational = rational.
F. \( \sqrt{18} + \sqrt{21} \)
- Both \( \sqrt{18} \) and \( \sqrt{21} \) are irrational → This is irrational + irrational; this doesn't demonstrate that the statement is false.
Thus, the expressions that show the original statement is false would be:
**A, B, C**.
A rational number is any number that can be expressed as the quotient of two integers, such as fractions or whole numbers. An irrational number cannot be expressed as a fraction of two integers.
Let's analyze the options:
A. \( \sqrt{25} + \pi \)
- \( \sqrt{25} = 5 \) (rational) + \( \pi \) (irrational) → This is rational + irrational = irrational.
B. \( 0.56 + \pi \)
- \( 0.56 \) (rational) + \( \pi \) (irrational) → This is rational + irrational = irrational.
C. \( \frac{7}{8} + \sqrt{13} \)
- \( \frac{7}{8} \) (rational) + \( \sqrt{13} \) (irrational) → This is rational + irrational = irrational.
D. \( \pi + \sqrt{17} \)
- \( \pi \) (irrational) + \( \sqrt{17} \) (irrational) → This is not a sum of rational + irrational; we can't determine if it's rational or irrational without further information.
E. \( 0.45 + 0.96 \)
- Both \( 0.45 \) and \( 0.96 \) are rational → This is rational + rational = rational.
F. \( \sqrt{18} + \sqrt{21} \)
- Both \( \sqrt{18} \) and \( \sqrt{21} \) are irrational → This is irrational + irrational; this doesn't demonstrate that the statement is false.
Thus, the expressions that show the original statement is false would be:
**A, B, C**.