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Prove that Sqrt3+Sqrt5 is an Irrational Number. - CBSE Class 9 - Mathematics

ConceptIrrational Numbers

Question

Prove that sqrt3+sqrt5 is an irrational number.

Solution

Given that sqrt3+sqrt5 is an irrational number

Now we have to prove sqrt3+sqrt5 is an irrational number

Let x=sqrt3+sqrt5 is a rational

Squaring on both sides

rArrx^2=(sqrt3+sqrt5)^2

rArrx^2=(sqrt3)^2+(sqrt5)^2+2sqrt3xxsqrt5

rArrx^2=3+5+2sqrt15

rArrx^2=8+2sqrt15

rArr(x^2-8)/2=sqrt15

Now  x is rational

⇒ x2 is rational

rArr(x^2-8)/2 is rational

rArr sqrt15 is rational

But, sqrt15 is an irrational

Thus we arrive at contradiction that sqrt3+sqrt5 is a rational which is wrong.

Hence sqrt3+sqrt5 is an irrational.

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Solution Prove that Sqrt3+Sqrt5 is an Irrational Number. Concept: Irrational Numbers.
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