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.
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No. The easiest counter-example to show that the product of two irrational numbers can be a rational number is that the product of √2 and √2 is 2. Likewise, the cube root of 2 is also an irrational number, but the product of 3√2, 3√2 and 3√2 is 2.
Not always. For example sqrt(2) and 1/sqrt(2) are both irrational, but their product is the rational number 1.
No, and I can prove it: -- The product of two rational numbers is always a rational number. -- If the two numbers happen to be the same number, then it's the square root of their product. -- Remember ... the product of two rational numbers is always a rational number. -- So the square of a rational number is always a rational number. -- So the square root of an irrational number can't be a rational number (because its square would be rational etc.).
They are not. Sometimes they are irrational. Irrational numbers cannot be expressed as a fraction.
A rational number in essence is any number that can be expressed as a fraction of integers (i.e. repeating decimal). Taking the product of any number of rational numbers will always yield another rational number.