Here is the question: Is the product of two irrational numbers always an irrational number? Justify your answer.
My answer is: No. The square root of two is an irrational number. If you multiply the square root of two by the square root of two, you get two which is a rational number. Sqrt. 2 * Sqrt 2= Sqrt.4= 2 Sqrt 2 * Sqrt 18 = Sqrt 36= 6 Is this correct? Thanks.
yes.
Yes, your answer is correct. The product of two irrational numbers is not always irrational. The examples you provided demonstrate that multiplying irrational numbers, such as the square root of 2, can result in a rational number (2) or an integer (6). Therefore, it can be concluded that the product of two irrational numbers can be irrational, rational, or an integer, depending on the specific irrational numbers being multiplied.
Your answer is partially correct. The product of two irrational numbers can be either rational or irrational depending on the numbers involved.
To further explain, let's consider the multiplication of two irrational numbers:
1) If you multiply two irrational numbers and the result is a rational number, it means that the irrational numbers were not truly irrational. In your example, multiplying √2 by √2 indeed results in 2, which is a rational number. This implies that √2 is not truly irrational, but rather a rational number in disguise. This is because the rational number 2 can be expressed as the ratio of two integers, which is not supposed to be the case for irrational numbers.
2) However, if you multiply two irrational numbers and the result is an irrational number, then the given statement is true. In your example, if you multiply √2 by √18, you get √36, which simplifies to 6. Since 6 is a rational number (6=6/1), your second example does not demonstrate that the product of two irrational numbers is always irrational.
In conclusion, the product of two irrational numbers may or may not be irrational. It depends on the specific irrational numbers being multiplied.