Question

prove that 3 + 2 root 5 is irrational number​

Answer 1

Step-by-step explanation:

to prove

$3 + 2 \sqrt{5}$

is irrational

let 3+2root 5 =p/q

root 5=p/q-3÷2

since LHS is irrational so RHS is also irrational

Answer 2

ANSWER:

GIVEN:

• Number = 3+2√5

TO PROVE:

• (3+2√5) is an Irrational number.

SOLUTION:

Let (3+2√5) be a rational number which can be expressed in the form of p/q where p and q have no common factor other than 1.

$\implies \: 3 + 2 \sqrt{5} = \dfrac{p}{q} \\ \\ \implies \: 2 \sqrt{5} = \dfrac{p}{q} - 3 \\ \\ \implies \: 2 \sqrt{5} = \dfrac{p - 3q}{q} \\ \\ \implies \: \sqrt{5} = \frac{p - 3q}{2q}$

Here:

• (p-3q)/2q is rational but √5 is an irrational.
• This our contradiction is wrong.
• So (3+2√5) is an irrational number.

NOTE:

• This method of proving an irrational number is called contradiction method.
• In contradiction method we first contradict a fact then we prove that our contradiction is wrong.