Question

Show that √2+√3 are irrational numbers.​

Answer 1

Let √2 + √3 = (a/b) is a rational no.

On squaring both sides , we get

2 + 3  + 2√6 = (a2/b2)

So,5 + 2√6 = (a2/b2) a rational no.

So, 2√6 = (a2/b2) – 5

Since, 2√6 is an irrational no. and (a2/b2) – 5 is a rational no.

So, my contradiction is wrong.

So, (√2 + √3) is an irrational no

Hope it helps!!

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Answer 2

ANSWER:

• √2+√3 is an Irrational number.

GIVEN:

• Number = √2+√3

TO PROVE:

• √2+√3 is an irrational number.

SOLUTION:

Let √2+√3 be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

=> √2+√3 = p/q

=> √2 = (p/q) -√3

Squaring both sides:

=> (√2)² = [ (p/q) -√3]²

=> 2 = p²/q² +3 -(2√3p)/q

=> (2√3p)/q = (p²/q²)+3-2

=> (2√3p)/q = (p²/q²) +1

=> (2√3p)/q = (p²+q²)/q²

=> 2√3p = (p²+q²)/q

=> √3 = (p²+q²)/2pq

Here:

• (p²+q²)/2pq is rational but √3 is Irrational.
• Thus our contradiction is Wrong.
• √2+√3 is an Irrational number.