Prove that
[tex]2 + \sqrt{5 \:} [/tex]
Is an irrational number
​

1 thought on “Prove that <br />[tex]2 + \sqrt{5 \:} [/tex]<br />Is an irrational number <br />​”

1. Answer:

   [tex] \: [/tex]

   Given: √2+√5

   We need to prove√2+√5 is an irrational number.

   Proof:

   Let us assume that √2+√5 is a rational number.

   A rational number can be written in the form of p/q where p,q are integers and q≠0

   √2+√5 = p/q

   On squaring both sides we get,

   (√2+√5)² = (p/q)²

   √2²+√5²+2(√5)(√2) = p²/q²

   2+5+2√10 = p²/q²

   7+2√10 = p²/q²

   2√10 = p²/q² – 7

   √10 = (p²-7q²)/2q

   p,q are integers then (p²-7q²)/2q is a rational number.

   Then √10 is also a rational number.

   But this contradicts the fact that √10 is an irrational number.

   Our assumption is incorrect

   √2+√5 is an irrational number.

   Hence proved..

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