Question

Prove that root5 + root7 is an irrational number.

Answer 1

To Prove :

• $\sf{ \sqrt{5} + \sqrt{7} \: is \: irrational}$

Let us assume that $\sqrt{5} + \sqrt{7}$ is a rational number, which let us take it as a.

Now,

⟶ $\sf{\sqrt{5} + \sqrt{7} = a}$

On squaring on both sides, we get :

⟶ $\sf{( \sqrt{5} + \sqrt{7})^{2} = {a}^{2} }$

⟶ $\sf{12 + 2 \sqrt{35} = {a}^{2} }$

⟶ $\sf{2 \sqrt{35} = {a}^{2} - 12}$

⟶ $\sf{ \sqrt{35} = \frac{ {a}^{2} - 12 }{2} }$

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$\sf{ \frac{ {a}^{2} - 12 }{2} }$ is a rational number, but $\sqrt{35}$ is irrational.

And we know that, One rational number can never be equal to an irrational number.

Which means that, our assumption was wrong.

⤐ $\sqrt{5} + \sqrt{7}$ is irrational.

$\sf{ \blue{Hence \: Proved!}}$

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Hnss lo thoda sa, itna mushkil bhi nahi hai yaar ye :yawn: xD xD