# Prove that ?7+3/2 is irrational

Dear Student,

Please find below the solution to the asked query:

Let  $\sqrt{7}$$\frac{3}{2}$ is a rational number .
Then we we can represent it in form of $\frac{p}{q}$ , where p and q are co-prime integers, So
$\sqrt{7}$$\frac{3}{2}$    =  $\frac{p}{q}$

$⇒$$\frac{p}{q}$  - ​$\sqrt{7}$     =  $\frac{3}{2}$

Now squaring both side ,we get

$⇒$$\frac{p}{q}$  - ​$\sqrt{7}$  )2${\left(\frac{3}{2}\right)}^{2}$

= $\frac{9}{4}$

$⇒$ $\frac{{p}^{2}}{{q}^{2}}$ + 7 - $\frac{9}{4}$ = 2$\sqrt{7}$ $\frac{p}{q}$

$⇒$$\frac{{p}^{2}}{{q}^{2}}$ +  $\frac{19}{4}$ = 2$\sqrt{7}$ $\frac{p}{q}$

= $\sqrt{7}$

$⇒$  = $\sqrt{7}$

Hence

$\sqrt{7}$    is a rational number  .

But we know that $\sqrt{7}$   is a irrational number , So that contradict fact that $\sqrt{7}$   is a irrational number .
So,
Our assumption is incorrect ,

Hence $\sqrt{7}$$\frac{3}{2}$   is a irrational number                                (  Hence proved )

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