Question

Prove
$3 + 2 \sqrt{5}$
is irrational?

Answer 1

heya ....

here is your answer↓

→ refer to the attachment

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hope it helps

Answer 2

let us suppose that 
3 + 2 \sqrt{5}3+2√​5​​​ 
is a rational no.

so,
3 + 2 \sqrt{5} = \frac{ {p}^{2} }{ {q}^{2} }3+2√​5​​​=​q​2​​​​p​2​​​​ 
(where p and q are co prime integers and q isn't equal to 0).

2 \sqrt{5} = \frac{ {p}^{2} }{ {q}^{2} } - 32√​5​​​=​q​2​​​​p​2​​​​−3 


\sqrt{5} = \frac{ {p}^{2} - 3 {q}^{2} }{2 {q}^{2} }√​5​​​=​2q​2​​​​p​2​​−3q​2​​​​ 
clearly it is rational number but sqrt 5 is an irrational no.
it isn't possible ..

So, our assumption was wrong..

Therefore, 
3 + 2 \sqrt{5} \: is \: an \: irrational \: number3+2√​5​​​isanirrationalnumber 

HOPE IT HELPS YOU '_'