Prove that✓5+✓3 is irrational.

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Prove that✓5+✓3 is irrational.

Posted by Banty Kumari 1 year, 9 months ago

CBSE > Class 10 > Mathematics

3 answers

Nikita Yadav 1 year, 9 months ago

To prove it let us assume it to be a rational number Rational numbers are the ones which can be expressed in p/q form where p,q are integers and q isn't equal to 0 √3+√5=p/q √3=(p/q)-√5 Squatting on both sides 3=p²/q²-(2√5p)/q+5 (2√5p)/q=5-3-p²/q² 2√5p/q=(2q²-p²)/q² √5=(2q²-p²)/q²*q/2p √5=(2q²-p²)/2pq As p and q are integers RHS is rational As RHS is rational LHS is also rational i.e √5 is rational But this contradicts the fact that √5 is irrational This contradiction arose because of our false assumption So √3+√5 is irrational

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Kulveer Kaur 1 year, 9 months ago

Here's your answer To prove: √3+√5 is irrational To prove it let us assume it to be a rational number Rational numbers are the ones which can be expressed in p/q form where p,q are integers and q isn't equal to 0 √3+√5=p/q √3=(p/q)-√5 Squatting on both sides 3=p²/q²-(2√5p)/q+5 (2√5p)/q=5-3-p²/q² 2√5p/q=(2q²-p²)/q² √5=(2q²-p²)/q²*q/2p √5=(2q²-p²)/2pq As p and q are integers RHS is rational As RHS is rational LHS is also rational i.e √5 is rational But this contradicts the fact that √5 is irrational This contradiction arose because of our false assumption So √3+√5 is irrational

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Tanu Bhardwaj 1 year, 9 months ago

First we let tha root5+ root3 = a upon b [ here a and be are composite no. Where a belongs integer and b is not equal to 0 Then on squaring both side it will we root5^2+root3^2 = a^2 upon b^2. Now it will be 8 = a^2 upon b^2 By cross multiply It will be 8b^2 = a^2 we make it (i) equation Let a=8k Then on squaring we put here a^2 value as 8b^2 8b^2 = 64k^2 on reducing it it will be... b^2 = 8k^2 we make it (ii) equation we see here from equ.(i) and (ii) That this contradict our assumption Therefore root 5 + root 3 is an irrational no. Hence proved!!...... 🥰🤗

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