Question

prove that
not any inappropriate answer ​

Answer 1

Answer:

see the attachment or your answer

Answer 2

To prove :

√5 - √3 is irrational numbers

Step-by-step explanation :

Let √5 - √3 be a rational number.

A rational number can be written in the form of p/q where p,q are integers.

$\sqrt{5} -\sqrt{3}=\frac{p}{q} \\\\ \sqrt{5}=\frac{p}{q} +\sqrt{3}$

Squaring on both sides,

$\sqrt{5} ^2 =(\frac{p}{q}+\sqrt{3})^2 \\\\\\ 5=\frac{p^2}{q^2}+\sqrt{3}^2+2(\frac{p}{q})(\sqrt{3}) \\\\\\ 5=\frac{p^2}{q^2}+3+\frac{2\sqrt{3}p}{q} \\\\\\ 5-3-\frac{p^2}{q^2}=\frac{2\sqrt{3}p}{q} \\\\\\ \frac{2\sqrt{3}p}{q}=2-\frac{p^2}{q^2} \\\\\\ \frac{2\sqrt{3}p }{q} =\frac{2q^2-p^2}{q^2} \\\\\\ 2\sqrt{3} p=\frac{2q^2-p^2}{q}\\\\\\\sqrt{3}=\frac{2q^2-p^2}{2pq}$

p,q are integers then (2q²-p²)/2pq is a rational number.

Then √3 is also a rational number.

But this contradicts the fact that √3 is an irrational number.

∴ Our supposition is false.

Hence, √5 - √3 is an irrational number.