Prove that√3+2√5 is a irrational number.
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Let us assume that √3+2√5 is rational number.
∴ √3+2√5=a/b where a and b are rational numbers.
squaring both sides,
(√3+2√5)²=a²/b²
⇒ 3+4√5+20=a²/b²
⇒23+4√5=a²/b²
⇒4√5=(a²/b²) -23
⇒√5=[(a²/b²)-23]/4
Now, it is clear that R.H.S. is rational but L.H.S. is irrational as√5 is irrational.
A rational number cannot be equal to an irrational number.
This contradicts our assumption.
Our assumption is wrong.
√3+2√5 is irrational.
HOPE IT HELPS YOU..
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