22. Which of the following conditions imply that the real number x is rational?
I. x1/2 is rational.
II. x2 and x5 are rational.
III. x2 and x4 are rational.
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A. I and II only |
B. I and III only |
C. II and III only |
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D. I, II, and III |
E. The correct answer is not given by A, B, C, or D. |
Solution: The answer is A
Condition I: Implies that x is a rational number.
The above statement is true because the set of rational numbers has the property that the product of any two rational numbers is also a rational number.
Condition II: Implies that x is a rational number.
The above statement is true because the set of rational numbers has the property that the quotient of any two rational numbers is also a rational number.
Condition II: Does not imply that x is a rational number.
If x2 and x4 are both rational then x may or may not be rational.
Example:
Which shows that x2 and x4 are both rational numbers. However, solving for x yields,
Since 21/2 is not a rational number, then x is not a rational number.