IDNLearn.com is your go-to resource for finding precise and accurate answers. Discover in-depth and trustworthy answers to all your questions from our experienced community members.
Sagot :
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
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
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.