To explain why A + B is irrational, we need to show that it cannot be expressed as a ratio of two integers.
Suppose that A + B is rational, which means we can write it as the ratio of two integers p and q (where q is not zero):
A + B = p/q
Now, we can substitute the values of A and B and simplify:
√363 + √27 = p/q
We can then rearrange the terms to isolate one of the square roots:
√363 = p/q - √27
We can square both sides of this equation to eliminate the square roots:
363 = p^2/q^2 + 27 - 2(p/q)√27
Notice that the right-hand side of this equation has a term with a square root. This means that if we assume that A + B is rational, we arrive at a contradiction: we have shown that √363 (which is equal to A) is irrational, which means that p^2/q^2 + 27 must also be irrational. However, the left-hand side of the equation is rational. Therefore, our assumption that A + B is rational must be false, and we conclude that A + B is irrational.
To explain why AB is rational, we can use the fact that the product of two rational numbers is rational.
We can rewrite A and B as follows:
A = √(363) = √(121 x 3) = √(11^2 x 3) = 11√3
B = √(27) = √(9 x 3) = √(3^2 x 3) = 3√3
Therefore, AB = 11√3 x 3√3 = 33 x 3 = 99.
Since 99 is a rational number (which can be expressed as the ratio of the integers 99 and 1), we conclude that AB is rational.