# 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.