Alright, let's determine if the number [tex]\(\pi \sqrt{4}\)[/tex] is rational or irrational.
First, let's simplify the given expression [tex]\(\pi \sqrt{4}\)[/tex]:
1. Calculate [tex]\(\sqrt{4}\)[/tex]:
- The square root of 4 is 2 because [tex]\(2 \times 2 = 4\)[/tex].
- Therefore, [tex]\(\sqrt{4} = 2\)[/tex].
2. Substitute [tex]\(\sqrt{4}\)[/tex] with 2 in the original expression:
- The expression becomes [tex]\(\pi \cdot 2\)[/tex] or [tex]\(2\pi\)[/tex].
Next, we need to determine the nature of [tex]\(2\pi\)[/tex]:
3. Recognize the properties of the numbers involved:
- [tex]\(\pi\)[/tex] (pi) is a well-known irrational number. An irrational number cannot be expressed as a simple fraction of two integers.
- The number 2 is a rational number, as it can be expressed as the ratio [tex]\(\frac{2}{1}\)[/tex].
4. Determine the product of a rational and an irrational number:
- Multiplying any rational number by an irrational number results in an irrational number. This is because the product cannot be expressed exactly as a fraction of two integers.
Putting this all together, the number [tex]\(2\pi\)[/tex] is an irrational number. Hence, [tex]\(\pi \sqrt{4}\)[/tex] is indeed an irrational number.
So, the correct answer is:
(B) Irrational
