To determine whether the expression [tex]\(\sqrt{8} \cdot 5\)[/tex] is rational or irrational, we need to examine the properties of the numbers involved.
1. Identify the nature of [tex]\(\sqrt{8}\)[/tex]:
- [tex]\(\sqrt{8}\)[/tex] is the square root of 8.
- 8 is not a perfect square (i.e., there is no integer [tex]\( n \)[/tex] such that [tex]\( n^2 = 8 \)[/tex]).
- Therefore, [tex]\(\sqrt{8}\)[/tex] cannot be expressed as a ratio of two integers, meaning [tex]\(\sqrt{8}\)[/tex] is an irrational number.
2. Analyze the multiplication of an irrational number by a rational number:
- The number 5 is a rational number because it can be expressed as the ratio [tex]\(\frac{5}{1}\)[/tex], where both 5 and 1 are integers.
- When an irrational number (such as [tex]\(\sqrt{8}\)[/tex]) is multiplied by a non-zero rational number (such as 5), the result is also an irrational number. This is due to the fact that the product cannot be expressed as a ratio of two integers.
In conclusion, [tex]\(\sqrt{8} \cdot 5\)[/tex] is the product of an irrational number and a rational number, which results in an irrational number. Thus, the expression [tex]\(\sqrt{8} \cdot 5\)[/tex] is:
Irrational
