Question

6. Prove that
$\sqrt{2} + \sqrt{5} \: is \: irrational, \: using \\ \: the \: fact \: that \: \sqrt{10} \: is \: irrational.$


Standard:- 10

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Answer 1

Given: $\sqrt{10}$ is irrational

To Prove: $\sqrt{2}+\sqrt{5}$ is irrational.
Proof:

Consider:

$\begin{array}{ccc}(\sqrt{2}+\sqrt{5})^2 & = & (\sqrt{2})^2+2(\sqrt{2})(\sqrt{5}) + (\sqrt{5})^2 \\ \\ & = & 2+2\sqrt{10}+5 \\ \\ & = & 7 + 2\sqrt{10} \end{array}$


Now, 

$\sqrt{10}$ is irrational. 

Also, 2 is a rational number.

We know that A product of a rational and an irrational number will always be an irrational number.


So, $2\sqrt{10}$ is irrational.


Also, The sum of a rational number and an irrational number is always an irrational number.

So, $7+2\sqrt{10}$ is an irrational number.

Also, since we have $(\sqrt{2}+\sqrt{5})^2=7+2\sqrt{10}$, we know that $(\sqrt{2}+\sqrt{5})^2$ is also an irrational number.


Also, The square-root of an irrational number is also going to be a rational number.


So, $\sqrt{(\sqrt{2}+\sqrt{5})^2} = (\sqrt{2}+\sqrt{5})$ is also irrational.



We have the following:


 

$\sqrt{10} \text{ is irrational} \\ \\ \implies 2\sqrt{10} \text{ is irrational} \\ \\ \implies 7+2\sqrt{10} \text{ is irrational} \\ \\ \implies (\sqrt{2}+\sqrt{5})^2 \text{ is irrational} \\ \\ \implies (\sqrt{2}+\sqrt{5}) \text{ is irrational} \\ \\ \text{Hence Proved}$

Answer 2

Hey there !

Solution :

Fact : √10 is irrational

To Prove : √ 2 + √ 5 is an irrational number

Proof :

Let us consider √ 2 + √ 5 as a rational number.

So squaring the number we get,

=> ( √ 2 + √ 5 )² = ( √ 2 )² + ( √ 5 )² + 2 × √ 2 × √ 5

=> ( √ 2 + √ 5 )² = 2 + 5 + 2 √10

=> ( √ 2 + √ 5 )² = 7 + 2 √10

We know that Rational number on squaring gives another rational number.

But we know the fact that √10 is irrational. 

Any rational number when added, subtracted, multiplied or divided by any irrational number yields us with an irrational number only.

Hence 2 × √10 = Irrational number ( Since they are multiplied )

Again 7 + 2 √10 is irrational number. ( Since they are added )

So it contradicts the fact that, ( √ 2 + √ 5 )² is a rational number. Hence
( √ 2 + √ 5 )² is an irrational number.

But we know the fact that, 

Any irrational number when are square rooted also give an irrational number.

Hence Square root of ( √ 2 + √ 5 )² is also an irrational number.

Square root of ( √ 2 + √ 5 )²  is ( √ 2 + √ 5 )

Hence ( √ 2 + √ 5 ) is also an irrational number.

Hence Proved !

Hope my answer helped :-)