Question

prove that (2+3√5) is an irrational number, given that √5 is an irrational numbe.​

Answer 1

GIVEN:

• 2 + 3√5

TO FIND:

• Prove that 2 + 3√5 is an Irrational number

SOLUTION:

Let 2+3√5 is a rational number

Now,

2+3√5 can be written in the form of p and q, where p and q are integers and q ≠ 0

According to given conditions:-

$\rm{\hookrightarrow 2 + 3\sqrt{5} = \dfrac{p}{q}}$

$\rm{\hookrightarrow 3\sqrt{5} = \dfrac{p}{q} - 2 }$

$\rm{\hookrightarrow 3\sqrt{5} = \dfrac{p-2q}{q}}$

$\rm{\hookrightarrow \sqrt{5} = \dfrac{p-2q}{3q}}$

We have given that, √5 is an irrational number,

So,

p–2q/3q is also an irrational number

This shows our assumption is incorrect.

So, 2 + 3√5 is an Irrational number.

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