Question

Prove that root P + root q is irrational where p and q are a primes

Answer 1

To ProvE :-

• √p + √q is a Irrational Number.

$\red{\frak{Given}}\begin{cases}\sf A \ number \ \sqrt{p} + \sqrt{q} .\\\\\textsf{ p and q are primes .} \end{cases}$

We need to prove that , √p + √q is a Irrational Number. So , on the contrary let us assume that it is a Rational number . So it can be expressed in the form of a/b where a and b are integers and b is not equal to zero. Also a & b are comprimes.

Therefore ,

$\sf\longrightarrow \sqrt{p}+\sqrt{q}=\dfrac{a}{b}$

On squaring both sides ,

$\sf\longrightarrow (\sqrt{p}+\sqrt{q})^2=\bigg(\dfrac{a}{b}\bigg) ^2\\\\\\\sf\longrightarrow p + q + 2\sqrt{pq} = \dfrac{a^2}{b^2} \\\\\\\sf\longrightarrow 2\sqrt{pq}= \dfrac{a^2}{b^2}-p-q \\\\\\\sf\longrightarrow \sqrt{pq}=\dfrac{a^2}{2b^2} -\dfrac{p}{2}-\dfrac{q}{2}$

Now the RHS term is a rational expression , since all p , q , a & b are Rational. And LHS term will be irrational since p are q are primes , and they are under square root. Since Rational ≠ Irrational . Therefore , our assumption was wrong .

$\sf\longrightarrow\red{ \sqrt{p}+\sqrt{q}= Irrational Number }$

Hence Proved !