Question

prove that is a irrational numbers

$\sqrt{5 + \sqrt{7} }$


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Answer 1

Answer:

Hence proved that the given $\sqrt{5+\sqrt{7}$ is an irrational number

To prove:

To prove whether $\sqrt{5+\sqrt{7}$ is irrational or not.

Solution:

Let us assume that

$\sqrt{5+\sqrt{7}$ be rational, and let p/q are co-prime where q is not equal to zero(0).

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

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

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

We know that $\sqrt{5}$ is irrational while p/q form is rational.

Hence it contradicts our assumption of  $\sqrt{5+\sqrt{7}$ is rational.

Hence, it is proved that $\sqrt{5+\sqrt{7}$ is irrational.