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.
1 answer