Question

prove that √3+√5 is irrational.​

Answer 1

Answer:

prove that √3+√5 is irrational.

Step-by-step explanation:

To prove : 3+5 is irrational.

Let us assume it to be a rational number.

 

Rational numbers are the ones that can be expressed in qp form where p,q are integers and q isn't equal to zero.

3+5=qp

3=qp−5

squaring on both sides,

 

3=q2p2−2.5(qp)+5

⇒q(25p)=5−3+(q2p2) 

⇒q(25p)=q22q2−p2

⇒5

I hope this answer will help you.....

Answer 2

Given:-

• $\rm{\sqrt{3} + \sqrt{5}}$

To Find:-

• To Prove that it us Irrational.

Now,

Let us assume that $\rm{\sqrt{3} + \sqrt{5}}$ is rational number equal to "x"

$\implies\rm{\sqrt{3} + \sqrt{5} = x}$

• Squaring both sides

$\implies\rm{(\sqrt{3} + \sqrt{5}})^2 = (\dfrac{P}{q})}$

$\implies\rm{(\sqrt{3})^2+ 2\times{\sqrt{3}}\times{\sqrt{5}} + (\sqrt{5})^2= x^2}$

$\implies\rm{ 3 + 2\sqrt{15} + 5 = x^2}$

$\implies\rm{ 8 + 2\sqrt{15} = x^2}$

$\implies\rm{ 2\sqrt{15} = x^2 - 8}$

$\implies\rm{\sqrt{15} = \dfrac{x^2 - 8}{2}}$

Therefore, if x is a Rational number then x² will also be the Rational number.

From here we get √15 is a Rational number but it contradicate the fact that √15 is a Irrational number

Hence, √3 + √5 is a Irrational number.