Question

prove that √5-√3 is irrational​ number

Answer 1

PROVE =

 Suppose that √5 -√3 is rational number say r.

then,

$\sqrt{5} -\sqrt{3}$ =r  ( note that r is not equal to 0 )

⇒-√3 = r -√5

squre  both  the  side

- √3 ² = (r - √5 )²

3 = r² + 5 - 2√5 r

2√5 r = r² -2

√5 = $\frac{r^{2}-2 }{2r}$

as ,

r is rational and r≠ 0

so $\frac{r^{2}-2 }{2r}$ is rational

⇒√5 is rational number.

but this contradict that √5 is irrational.

hence

our supposition is wrong

therefor

√5 -√3 is irrational.