Question

If√5 is irrational , prove 2+3√5 irrational

Answer 1

let's

2 + 3√5 be a rational no.

so , 2 + √5 = a/b

√5 = a/b - 2

( a/b - 2 ) is an rational no .

so, √5 should be a rational no.

A fact has generated √5 is irrerational no

our asuption is wrong .

so , 2+√5 is an irrerational no

Answer 2

$\displaystyle\huge\red{\underline{\underline{ANSWER}}}$

Given:3 + 2√5

To prove:3 + 2√5 is an irrational number.

Proof:

Letus assume that 3 + 2√5 is a rational number.

Soit can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.

so it contradictsour assumption.

Our assumption of3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved