Question

Show root 2 plus root 5 as irrational by showing root 10 irritation​

Answer 1

Answer:

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Show

$\sqrt{2 + \: \sqrt{5} }$

By showing

$\sqrt{10}$

Irrational

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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:

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