Question

Prove that
$3 + 2 \sqrt{5}$
is irrational.(Without calculating directly)

Answer 1

heya mate...here's ur answer
let 3 + 2 root 5 is rational 
therefore 3 + 2 root 5=a/b(where a and b are co prime integer ) 
root 5=a-3/2b 
because a-3/2b is rational 
therefore root 5 will also be rational 
but we know that, 
root 5 is irrational 
that is our assumption is wrong 
that is 3+2 root 5 is irrational..

hope it helps uh..anyway
wishing u a beautifull night

Answer 2

Hey.....

At first we know that √5 is an irrational no...

And Now , let 3 +2√5 is a rational no...

3+2√5-3=2√5--------(1)..
{rational -rational =rational}

AQ to eq. (1)2√5 is a rational so...

2√5×1/2=√5---------(2)
{rational × rational = rational}

A/Q to eq. (2) √5 is a rational no..

But we know that √5 is an irrational so, our contradict was wrong and...

2+3√5 is an irrational no....

Hope this will help u......✌✌✌✌✌.....