How are irrational numbers the result of adding rational numbers and irrational numbers or multiplying non zero rational numbers and irrational numbers

Rational numbers are numbers that can be expressed as a fraction (ratio). Irrational numbers can not be expressed like that (like sqrt(2)).

To prove your statement, assume the opposite until you have a contradiction.

If the result of adding them would be rational, then your irrational number can be expressed as the difference of two rational numbers, which itself is also a rational number. That cannot be, because it should be an irrational number. This contradiction forces that rational + irrational = irrational.

You can reason the same way for multiplication. Suppose rational * irrational = rational, you find that your irrational can be expressed as the fration of two rationals, which is a contradiction.