Question

prove that following is irrational
$5 + 3 \sqrt{2}$
​

Answer 1

Answer:

Step-by-step explanation:

We have to prove 5+3  

2

​  

 is an

irrational number

So let 5+3  

2

​  

 is a rational number

So  

q

p

​  

=5+3  

2

​  

 where gcd(p.q)=1,Rq



=0

⇒p=5q+3q  

2

​  

 

⇒p−5q=3q  

2

​  

 

⇒  

3q

p−5q

​  

=  

2

​  

 

⇒  

3q

p

​  

−  

3q

5q

​  

=  

2

​  

 

⇒  

3q

p

​  

−  

3

5

​  

=  

2

​  

...(1)

Here in eq (1)  

3

5

​  

 is a rational no and

in case of  

3q

p

​  

 there is two cases

Case I :-

let 3 divides p then 3r=p

So  

3q

3r

​  

⇒  

q

r

​  

 

Here gcd (p,r) = 1 Hence gcd (r,q) = 1

because r is a divisor of q so.

3q

p

​  

 is a rational number.

In case II :-

let 3 divides not divide p

hence gcd of (p,3q) = 1 because

3 is prime so,  

3q

p

​  

  is an rational no.

so. from eq (1) we can see that

both term are rational then whole

LHS is rational. By this we proved

that  

2

​  

 is a rational but this is

not true.

So, 5+3  

2

​  

 is an irrational no