given that root 2 is rational number prove that 5 + 3 root 2 is an irrational number
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Let us assume that 5+2√3 is rational
5+2√3 = p/q ( where p and q are co prime)
2√3 = p/q-5
2√3 = p-5q/q
√3 = p-5q/2q
now p , 5 , 2 and q are integers
∴ p-5q/2q is rational
∴ √3 is rational
but we know that √3 is irrational . This is a contradiction which has arisen due to our wrong assumption.
∴ 5+2√3 is irrational
5+2√3 = p/q ( where p and q are co prime)
2√3 = p/q-5
2√3 = p-5q/q
√3 = p-5q/2q
now p , 5 , 2 and q are integers
∴ p-5q/2q is rational
∴ √3 is rational
but we know that √3 is irrational . This is a contradiction which has arisen due to our wrong assumption.
∴ 5+2√3 is irrational
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