Question

Prove that the following numbers are irrational
(0) √2 + √5​

Answer 1

Answer: Use proof by contradiction. Assume that the sum is rationial, that is

2–√+5–√=ab

where a and b are integers with b≠0. Now rewrite this as

5–√=ab−2–√.

Squaring both sides of this equation we obtain

5=a2b2−22–√ab+2.

Now, carefully solve for 2–√ and obtain

2–√=−3b2a+a2b.

This implies that 2–√ is a rational number which is a contradiction. Thus

2–√+5–√

is an irrational number.

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