¶¶¶ 3 + 2√5 = a/ b

2√5 =a/b -3

√5 =a-3b/2b

√5 is rational....

Proof that 3/2√5 is irrational

Assumption: Let's assume that 3/2√5 is rational.

Definition: A rational number is a number that can be expressed as a ratio of two integers.

Let: Let 3/2√5 be expressed as p/q where p and q are integers and q ≠ 0.

Simplifying the equation:
- 3/2√5 = p/q
- 3q = 2p√5
- Squaring both sides, we get
- 9q^2 = 20p^2

Conclusion: Since 20p^2 is even, 9q^2 must also be even. Thus, q^2 must be even, which means q must be even.

Further simplifying the equation:
- 9q^2 = 20p^2
- Dividing both sides by 4, we get
- (9/4)q^2 = 5p^2/2

Conclusion: This means that p^2 must be even, which means p must also be even.

Contradiction: However, this contradicts our assumption that p and q have no common factors. If p and q are both even, then they have a common factor of 2.

Conclusion: Therefore, our assumption that 3/2√5 is rational is false. Hence, 3/2√5 is irrational.

Final Note: The proof can be generalized for any number of the form a/b√n, where a, b, and n are integers and n is not a perfect square.