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Examine, Whether the Following Number Are Rational Or Irrational: `Sqrt3+Sqrt5` - Mathematics

Examine, whether the following number are rational or irrational:

`sqrt3+sqrt5`

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Solution

Let `x=sqrt3+sqrt5` be the rational number

Squaring on both sides, we get

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

`rArrx^2=3+5+2sqrt15`

`rArrx^2=8+2sqrt15`

`rArrx^2-8=2sqrt15`

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

Now, x is rational number

⇒ x2 is rational number

⇒ x2 - 8 is rational number

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

`rArrsqrt15` is rational number

But `sqrt15` is an irrational number

So, we arrive at a contradiction

Hence `sqrt3+sqrt5` is an irrational number.

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 1 Number Systems
Exercise 1.4 | Q 3.05 | Page 31
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