Question

Prove that 5 + √3 is irrational.​

Answer 1

Step-by-step explanation:

Let us assume the given number be rational and we will write the given number in p/q form

⇒5-✓3=p/q

✓3=(5q-p)/q

We observe that LHS is irrational and RHS is rational, which is not possible.

This is contradiction.

Hence our assumption that given number is rational is false

⇒5− ✓3 is irrational

Answer 2

$\sf{\huge{\underline{\green{Answer :-}}}}$

Let us consider, 5 + √3 is a rational number.

The numbers a & b ( b≠ 0 )

5 + √3 = $\dfrac{a}{b}$

So, 5 + $\dfrac{a}{b}$ = √3

So we get, $\dfrac{5b}{b}$ + $\dfrac{a}{b}$ = √3 = √3

Since a & b are integers, we get 5b + a/b is rational,

So, √3 will also be a rational number.

We know, √3 is an irrational number

It is a contradiction in our statement.

Hence, 5 + √3 is irrational number.

______________________________________