Prove that 3 + 2 √5 is irrational.

Prove that 3 + 2 √5 is irrational.
Ans 1. let 3 + 2 √5 is irrational.
Therefore, we can find two integers a, b (b ≠ 0) such that
3 + 2 √5 = a/b
2 √5 = a/b – 3
√5 = 1/2 (a/b – 3)
Since a and b are integers, ½ (a/b – 3) will also be rational and
Therefore √5 is rational.
This contradicts the fact that √5 is irrational. Hence, our assumption
That 3 + 2√5 is rational is false. Therefore, 3 + 2 √5 is irrational.

Ques. Prove that the following are irrationals.
I) 1/ √2
Ans (I) Let 1/ √2 is rational.
Therefore, we can find two integers a, b (b ≠ 0) such that
1/√2 = a/b
√2 = b/a
b/a is rational as a and b are integers.
Therefore, √2 is rational which contradicts to the fact that √2 is irrational.
Hence, our assumption is false and 1/√2 is irrational.

II) 7√5
Ans (II). Let 7√5 is rational.
Therefore, we can find two integers a, b (b ≠ 0) such that
7√5 = a/b for some integers a and b
Therefore, √5 = a/ 7b
a/7b is rational as and b are integers
Therefore, √5 should be rational.
This contradicts the fact that √5 is irrational. Therefore, our assumption that 7√5 is rational is false. Hence, 7√5 is irrational.

III) 6 + √2
Ans (III). Let 6 + √2 be rational.
Therefore, we can find two integers a, b (b ≠ 0) such that
6 + √2 = a/b
√2 = a/b – 6
Since a and b are integers, a/b – 6 is also rational and hence, √2 should be rational. This contradicts the fact that √2 is irrational. Therefore, our assumption is false and hence, 6 + √2 is irrational.