Question

Prove that (root5+root3)^2 is irrational

Answer 1

(√5 + √3)²

__________ [ GIVEN ]

• We have to prove it irrational.

____________________________

Let us assume that (√5 + √3)² is rational number.

→ (√5 + √3)² = a/b

Here a and b are co-prime numbers

We know that

(a + b)² = a² + b² + 2ab

So,

→ (√5)² + (√3)² + 2(√5)(√3) = a/b

→ 5 + 3 + 2√15 = a/b

→ 8 + 2√15 = a/b

→ 2√15 = (a/b) - 8

→ 2√15 = (a - 8b)/b

→ √15 = (a - 8b)/2b

Here.. (a - 8b)/2b is rational number

So, √15 is also a irrational number.

But we know that √15 is irrational number.

So, our assumption is wrong.

(√5 + √3)² is irrational number.

_______ [ PROVED ]

___________________________