To find the two numbers that, when multiplied by the square root of 360, give a rational number, we can start by expressing 360 as a product of its prime factors:
360 = 2^3 * 3^2 * 5
Then, we can take the square root of 360:
√360 = √(2^3 * 3^2 * 5) = 2 * 3 * √5
Now, we can see that if we multiply 2 or 3 by √360, we will get a rational number because both 2 and 3 are rational numbers. On the other hand, if we multiply √5 by √360, we will get an irrational number because √5 is an irrational number.
Therefore, the two numbers that, when multiplied by √360, give a rational number are 2 and 3. These numbers are rational.