Which statements are true?
Options
- AThe square root of a positive number greater than 1 is less than the number.
- BThe square root of a positive number is always less than half of the number itself.
- CThe square root of a positive number less than 1 is less than the number.
- DThe square root of a positive number is less than 1.
- EThe square root of a perfect square is not a whole number.
- FThe square root of a positive number that is less than 1 is between 0 and 1.
Answer
The true statements are: 'The square root of a positive number greater than 1 is less than the number' and 'The square root of a positive number that is less than 1 is between 0 and 1'. These align with the fundamental properties of square roots for values in these ranges.
Explanation
To determine which statements are true, we must evaluate each against the mathematical properties of square roots: 1. The square root of a positive number greater than 1 is less than the number: True. For example, √4 = 2, and 2 < 4. 2. The square root of a positive number is always less than half of the number itself: False. For example, √1 = 1, which is greater than 0.5. Also, √0.25 = 0.5, which is greater than 0.125. 3. The square root of a positive number less than 1 is less than the number: False. In fact, it is greater. For example, √0.25 = 0.5, and 0.5 > 0.25. 4. The square root of a positive number is less than 1: False. This only applies to numbers between 0 and 1. √9 = 3, which is not less than 1. 5. The square root of a perfect square is not a whole number: False. By definition, a perfect square is the square of a whole number (e.g., √25 = 5). 6. The square root of a positive number that is less than 1 is between 0 and 1: True. If 0 < x < 1, then 0 < √x < 1. For instance, √0.49 = 0.7. In summary, only the first and last statements are mathematically sound properties of square roots.