Find each product. Then determine which conclusion can be drawn based on the results.
[tex]
\begin{array}{l}
\sqrt{2} \cdot \sqrt{2} \\
\sqrt{5} \cdot \sqrt{7} \\
\sqrt{2} \cdot \sqrt{18} \\
\sqrt{2} \cdot \sqrt{6}
\end{array}
[/tex]
A. The product of two irrational numbers can be rational or irrational.
B. The product of two irrational numbers cannot be determined.
C. The product of two irrational numbers is always rational.
D. The product of two irrational numbers is always irrational.
Answer (1)
Calculate 2 ⋅ 2 = 2 , which is rational.
Calculate 5 ⋅ 7 = 35 , which is irrational.
Calculate 2 ⋅ 18 = 6 , which is rational.
Calculate 2 ⋅ 6 = 12 , which is irrational.
The product of two irrational numbers can be rational or irrational. The product of two irrational numbers can be rational or irrational.
Explanation
Problem Analysis We are given four products to compute and analyze:
2 ⋅ 2
5 ⋅ 7
2 ⋅ 18
2 ⋅ 6
Our goal is to determine whether the product of two irrational numbers is always rational, always irrational, or can be either rational or irrational.
Calculations Let's compute each product:
2 ⋅ 2 = 2 . This is a rational number.
5 ⋅ 7 = 35 ≈ 5.916 . This is an irrational number.
2 ⋅ 18 = 36 = 6 . This is a rational number.
2 ⋅ 6 = 12 = 2 3 ≈ 3.464 . This is an irrational number.
Conclusion Now, let's analyze the results. We have:
The product of 2 and 2 is 2 , which is rational.
The product of 5 and 7 is 35 , which is irrational.
The product of 2 and 18 is 6 , which is rational.
The product of 2 and 6 is 12 , which is irrational.
Since we have examples where the product of two irrational numbers is rational (e.g., 2 ⋅ 2 = 2 ) and examples where the product is irrational (e.g., 5 ⋅ 7 = 35 ), we can conclude that the product of two irrational numbers can be either rational or irrational.
Final Answer Therefore, the correct conclusion is:
The product of two irrational numbers can be rational or irrational.
Examples
Understanding whether the product of two irrational numbers can be rational or irrational is useful in various fields, such as physics and engineering, where irrational numbers often arise in calculations involving lengths, areas, and volumes. For example, when calculating the area of a circle with an irrational radius, the result can sometimes be a rational number if it involves multiplying by another irrational number like π . This knowledge helps in simplifying complex expressions and understanding the nature of the results obtained in practical applications.
