Question

show that √3 + √5 rational √p + √q irrational √1 where √2 is irrational​

Answer 1

Step-by-step explanation:

To prove : √3+√5 is irrational.

Let us assume it to be a rational number.

Rational numbers are the ones that can be expressed in q/pform where p,q are integers and q isn't equal to zero.

√3 +√5= q/p

√3=q/p−√5

squaring on both sides,

3= q²/p²−2.√5 (q/p )+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 also 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 irrational.

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