Question

Given that root3 is airrational prove that 5- 2root3 is a an irrational number​

Answer 1

Answer:

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Step-by-step explanation:

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

Answer 2

Step-by-step explanation:

let 5-2√3 be a rational number

let 5-2√3 =p/q , where p&q are coprimes and q is not equal to 0

5-2√3=p/q

-2√3=(p/q)-5

-2√3=(p-5q)/q

2√3=(5q-p)/q

√3=(5q-p)/2q

here √3 is an irrational number i.e., given

and (5q-p)/2q is a rational no.

but any irrational no. can't be equal to any rational no.

therefore by contradiction our assumption is wrong

therefore 5-2√3 is an irrational number