If√5 is irrational , prove 2+3√5 irrational
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Answered by
35
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
Answered by
22
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
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