3. Prove indirectly: "It is known that √7 and √11 are irrational. Prove that √11-√7 is irrational."

Prove that if a is rational and b is irrational, then a+b is irrational.

Prove that is irrational.

Prove that 23 is not a rational number.

Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.

When can a theorem be cited as a reason reason in a proof?

15. Assume that are in and with and invertible. Solve for.