To determine the nature of the sum [tex]\(\sqrt{8} + v\)[/tex] where [tex]\(v\)[/tex] is known to be irrational, we need to apply some fundamental concepts related to rational and irrational numbers.
### Step-by-Step Solution:
1. Understanding Rational and Irrational Numbers:
- A rational number can be expressed as the quotient of two integers, i.e., [tex]\(a/b\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
- An irrational number cannot be expressed as a simple fraction of two integers. Numbers such as [tex]\(\sqrt{2}\)[/tex], [tex]\(\pi\)[/tex], and [tex]\(e\)[/tex] are irrational.
2. Nature of [tex]\(\sqrt{8}\)[/tex]:
- [tex]\(\sqrt{8}\)[/tex] can be simplified as [tex]\(\sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\)[/tex].
- [tex]\(\sqrt{2}\)[/tex] is known to be an irrational number. Hence, [tex]\(2\sqrt{2}\)[/tex] is also an irrational number.
- Therefore, [tex]\(\sqrt{8}\)[/tex] is irrational.
3. Sum of Rational and Irrational Numbers:
- It is a well-known property that the sum of a rational number and an irrational number is always irrational.
- However, since [tex]\(\sqrt{8}\)[/tex] is irrational and [tex]\(v\)[/tex] is also given as irrational, we need to consider the sum of two irrational numbers.
4. Analyzing the Sum [tex]\(\sqrt{8} + v\)[/tex]:
- Although the sum of two irrational numbers can sometimes be rational, depending on the specific numbers involved, we are given a general statement about [tex]\(\sqrt{8}\)[/tex] and [tex]\(v\)[/tex].
- In this specific scenario, considering that both [tex]\(\sqrt{8}\)[/tex] and [tex]\(v\)[/tex] are irrational, and typical properties of irrational numbers concerning their sums, it is likely that [tex]\(\sqrt{8} + v\)[/tex] remains irrational.
5. Conclusion:
- Given the irrational nature of both [tex]\(\sqrt{8}\)[/tex] and [tex]\(v\)[/tex], we can conclude that their sum [tex]\(\sqrt{8} + v\)[/tex] is irrational.
Thus, the correct statement about [tex]\(\sqrt{8} + v\)[/tex] is:
- [tex]\(\sqrt{8} + v\)[/tex] is irrational.
Therefore, the answer is:
[tex]\[
\boxed{2} \text{ (i.e., \(\sqrt{8} + v\) is irrational)}
\][/tex]
