Question

prove that √5-√2 is an irrational number​

Answer 1

Let us assume to the contray that √5 - √2  is a rational number.

Now,

A rational number can be written in the form of p/q where p,q are integers and q≠0.

⇒ √5 - √2 = p/q

On squaring both sides, we get;

2+5-2√10  = $\frac{p^{2} }{q^{2} }$

2√10 = $\frac{p^{2} }{q^{2} }$ - 7

√10 = $\frac{p^{2}-7q^{2} }{2q^{2} }$

∵ p,q are integers

∴ √10 must be rational

But, this contradicts the fact that √10 is an irrational number.

So, or assumption was wrong.

Therefore,  √5-√2 is an irrational number ​■

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                                                                               HENCE, PROVED