. Which of the following numbers has irrational square root? * 64 25 0.1
The Correct Answer and Explanation is:
The number that has an irrational square root among the options is 0.1.
Here’s why:
- 64:
- The square root of 64 is 8, because 64=8\sqrt{64} = 864=8. This is a rational number, as it can be expressed as a fraction (8/1).
- 25:
- The square root of 25 is 5, because 25=5\sqrt{25} = 525=5. This is also a rational number, as it can be expressed as a fraction (5/1).
- 0.1:
- The square root of 0.1 is approximately 0.316227766…, which is a non-repeating, non-terminating decimal. This is an example of an irrational number. Irrational numbers cannot be written as exact fractions of integers, and their decimal expansion goes on forever without repeating.
What makes 0.1’s square root irrational?
The key to identifying irrational numbers is whether their square root results in a non-repeating and non-terminating decimal. In this case, 0.1\sqrt{0.1}0.1 has a decimal expansion that neither terminates nor repeats, which satisfies the definition of an irrational number.
On the other hand, numbers like 64 and 25 have square roots that are finite integers, which are rational. Therefore, among the options, 0.1 is the number whose square root is irrational.
In summary:
- 64=8\sqrt{64} = 864=8, a rational number.
- 25=5\sqrt{25} = 525=5, a rational number.
- 0.1≈0.316227766…\sqrt{0.1} \approx 0.316227766…0.1≈0.316227766…, an irrational number.
