Question

prove that root 5 + root 7 is an irrational no.

Answer 1

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Answer 2

Answer:

Given : Number $\sqrt{5}+\sqrt{7}$

To find : Prove that the given number is an irrational number?

Solution :

Let  $\sqrt{5}+\sqrt{7}$ is a rational number

Where p/q are co-prime and q≠0

$\sqrt{5}+\sqrt{7}=\frac{p}{q}$

$\sqrt{5}=\frac{p}{q}-\sqrt{7}$

Squaring both sides,

$5=(\frac{p}{q})^2+7-2\sqrt{7}(\frac{p}{q})$

$2+\frac{p^2}{q^2}+7=2\sqrt{7}(\frac{p}{q})$

$\frac{2+\frac{p^2}{q^2}+7}{2(\frac{p}{q})}=\sqrt{7}$

Since, $\sqrt{7}$ is rational is a contradiction.

Therefore, our assumption is wrong.

$\sqrt{5}+\sqrt{7}$ is an irrational number.