Question

if
$\sqrt{yz}$
is an irrational number, prove that(√y+√z) is an irrational number.​

Answer 1

Step-by-step explanation:

I have proved in earlier exercises of this book that 2–√ and 3–√ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, 2–√+3–√ is irrational. My first question is, is this reasoning correct?

Secondly, the book wants me to use the fact that if n is an integer that is not a perfect square, then n−−√ is irrational. This means that 6–√ is irrational. How are we to use this fact? Can we reason as follows:

6–√ is irrational

⇒2⋅3−−−√ is irrational.

⇒2–√⋅3–√ is irrational

⇒2–√ or 3–√ or both are irrational.

⇒2–√+3–√ is irrational.